Spatial overlap is irrelevant for combining effects

In the DMG section on combining game effects¹:

Different game features can affect a target at the same time. But when two or more game features have the same name, only the effects of one of them—the most potent one—apply while the durations of the effects overlap. For example, if a target is ignited by a fire elemental’s Fire Form trait, the ongoing fire damage doesn’t increase if the burning target is subjected to that trait again. Game features include spells, class features, feats, racial traits, monster abilities, and magic items.

This section says nothing about whether or not 2 effects are spatially overlapped. The rule applies regardless of whether two moonbeam spells cover overlapping areas or not.

Creature size is irrelevant for combining effects

If a creature is larger than medium, its space includes multiple 5-foot squares. This creates the opportunity for a single creature to be simultaneously subjected to multiple non-overlapping areas of effect, which is a bit non-intuitive. However, nothing about the rules really changes in this situation. If a creature is subjected to two effects, then those effects combine according to the above rule, as normal, even if their areas don't overlap. In particular, if the two effects are ongoing and have the same name, the creature only suffers the effects of one of them.

A creature can only take damage from moonbeam once per turn

Instead of 2 non-overlapping moonbeam spells, let's set up a line of 10 moonbeams. Then let's cast haste on a giant eagle and send it flying through the entire line. How much damage does it take? Just 2d10 damage, halved on successful save. This is because the damage from moonbeam can only trigger once per turn (emphasis added):

When a creature enters the spell’s area for the first time on a turn or starts its turn there, it is engulfed in ghostly flames that cause searing pain, and it must make a Constitution saving throw. It takes 2d10 radiant damage on a failed save, or half as much damage on a successful one.

It doesn't matter that our eagle is entering multiple moonbeam spells. The eagle takes the damage the first time it enters any moonbeam spell on its turn, and no effects occur on any subsequent entries into any moonbeam spell for the remainder of that turn, because all of these spells' damage-dealing effects are ongoing effects from spells of the same name with overlapping durations, so they don't combine. Note that this is a logical outcome, because the exact same thing would happen if the eagle flew back and forth through a single moonbeam 10 times. Also note that the result is the same whether or not any of the 10 spells have overlapping areas.

Yes, there are weird edge cases, but that's OK

Instead of having 10 moonbeam spells going at once, let's have 10 druids each ready a moonbeam spell to release right in front of the eagle's flight path. Each one releases the spell right before the eagle reaches their targeted point, then drops their concentration as soon as the eagle takes damage from their spell, before the next druid releases their spell a split second later. Now, none of the spells' durations overlap, which means the above rule about combining effects with the same name technically no longer applies, and the eagle suffers the effects of entering a moonbeam spell 10 times in a single turn. Whoops.

Of course, it is utterly illogical and rather unsatisfying that this situation should produce a different result than an eagle flying through 10 ongoing moonbeams. But I'd argue that this isn't a problem in practice, because this illogical situation is never going to occur in actual play. It's probably never going to occur even with just 2 moonbeam spells. This would require 2 druids on the battlefield, one with an active moonbeam spell and one with a readied moonbeam spell, and then a creature would, in the span of one turn, need to run through one moonbeam spell, break the druid's concentration on that spell, trigger the other druid's readied action, and then intentionally enter the 2nd moonbeam spell's area.

Remember this important quote from the DMG's introduction:

The rules don’t account for every possible situation that might arise during a typical D&D session.

If such an exceptional case as the one described above actually arises in your game, the rules expect the DM to make an appropriate ruling. In this case, it would be reasonable to rule that the eagle only suffers the effects of the first moonbeam spell, just like in the prior example of 10 ongoing moonbeams.

Another edge case occurs when the moonbeam spells are each cast at different levels and/or by casters with different spell save DCs. Once again, the rules don't explain what to do in this edge case, and it's the DM's job to fill in the blanks. If the first moonbeam spell has a DC of 17 and deals 2d10 damage while the 2nd one has a DC of 15 and deals 3d10 damage, what happens when the eagle flies through both? I dunno. Ask your DM.

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¹Thanks to user Scott Dunnington for pointing out that the DMG's version of the rule for combining effects specifically calls out ongoing damage effects as affected by the rule, even though each individual instance of the damage is instantaneous.

The creature takes damage from both effects

(based on an optional rule)

However the area is only lit by dim light by one of the beams. This is because moon beam has two parts:

Until the spell ends, dim light fills the cylinder.

This is a continuous effect and the duration of that effect overlaps for both moonbeams, therefore the area is only lit by dim light once.

The second part of moonbeam says:

When a creature enters the spell’s area for the first time on a turn or starts its turn there, it is engulfed in ghostly flames that cause searing pain, and it must make a Constitution saving throw. It takes 2d10 radiant damage on a failed save, or half as much damage on a successful one.

This is an instantaneous effect. This effect triggers for both moonbeams at the start of the turn of the creature or when it enters the area. Now the question is, do they get resolved at the same time or at different times? If they would be resolved at the same time, the durations would match and only one would apply.

To resolve this question, we turn to Xanathars Guide to Everything. Xanathars guide on page 77 states (this is an optional rule, as most rules in Xanathars are mere suggestions):

In rare cases, effects can happen at the same time, especially at the start or end of a creature's turn. If two or more things happen at the same time on a character or monster's turn, the person at the game table - whether player or DM - who controls that creature decides the order in which those things happen.

If the two effects happen in some order, that means they happen after one and another. Then the duration clearly no longer overlaps, so both spells do damage to the creature.

(This has also been referenced by this highly relevant answer: https://rpg.stackexchange.com/a/128320/51849)

The creature takes damage from both

Everybody is quoting the DMG on spell effects, but that is completely irrelevant here because nobody is trying to overlap moonbeams over each-other.

If the question was 'can I cast two Moonbeams on top of each other and have a creature take damage twice?', the answer is, indeed, no, because you can't overlap spell effects, so that 'tile' will only be affected by the stronger of the two Moonbeams.

However, we're not overlapping Moonbeams; we have two separate instances of Moonbeam side-by-side, and by virtue of being Large, the creature happens to be in both of them at the same time.

Ask yourself this:

If a Medium-sized creature walked through both Moonbeams to pick up items placed in the center of each, would it take damage once or twice? I think everybody would agree that it'd take damage twice, because it passed through two separate moonbeams.

The Large creature has the exact same problem, except it's going through both of them at the same time because of its size.

The moment you start ruling that stepping into two moonbeams at the same time counts as 'the same effect', you're giving Large creatures the possibility of cheesing out less damage by simply positioning themselves advantageously.

If I stand here, I'm inside two moonbeams at the same time, I'll only take damage once, and I can then still decide which way to go without having to take the damage again, unlike my smaller friends who will have to decide which Moonbeam they want to stand in, and they might take damage from the other Moonbeam as well if they decide they need to go the other way after all.

Size doesn't matter; you cannot stack the same spell on the same target within its duration.

Let's compare PHB ("Combining Magical Effects"):

The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don't combine, however. Instead, the most potent effect--such as the highest bonus--from those castings applies while their durations overlap, or the most recent effect applies if the castings are equally potent and their durations overlap.

to the DMG ("Combining Game Effects"):

Different game features can affect a target at the same time. But when two or more game features have the same name, only the effects of one of them—the most potent one—apply while the durations of the effects overlap. For example, if a target is ignited by a fire elemental’s Fire Form trait, the ongoing fire damage doesn’t increase if the burning target is subjected to that trait again.

The PHB version is fairly clear, but the DMG Version is extremely clear and even gives an example that works in this case: ongoing damage doesn't increase if the effect is used twice. The descriptions don't say the spells need to affect the same area, but only that they affect the same target. Nothing here prevents two wizards from casting magic missile (as an example) on their turns and having them both deal damage.

Let's consider the implications of this. Could you cast 4 Incendiary Clouds at 4 corners of an Ancient Dragon and force 4 saves and multiple damages? No, you cannot. The designers have considered this and prevented it via the above rule.

Two castings of the same spell within their durations will not multiply damage, saving throws, or anything else.

The creature takes damage from both spells.

Moonbeam is cast on an area, not a creature. It then causes damage to creatures within that area. The effect of said damage is instantaneous, and so the rules for overlapping spells doesn't apply.

This is combining the same spell multiple times.

The rule you're referencing is found in chapter 10 of the PHB, Spellcasting, in the subsection Combining Magical Effects. [emphasis mine]

The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don't combine, however. Instead, the most potent effect — such as the highest bonus — from those castings applies while their durations overlap, or the most recent effect applies if the castings are equally potent and their durations overlap.

For example, if two clerics cast bless on the same target, that character gains the spell's benefit only once; he or she doesn't get to roll two bonus dice.

Or in our case, a creature under the effect of moonbeam would be affected by the more powerful of the two.

The relevant portion of moonbeam says this:

When a creature enters the spell's area for the first time on a turn or starts its turn there, it is engulfed in ghostly flames that cause searing pain, and it must make a Constitution saving throw. It takes 2d10 radiant damage on a failed save, or half as much damage on a successful one.

Clearly, the creature itself is the target of the spell, and not some portion of the creature. Moonbeam targets a location to set the spell's area of effect, as well as each creature it harms.