Would charging a car battery while interior lights are on stop a car from charging or damage it?

You probably did no damage. Most chargers have built-in protection for excessive amperage drain. Courtesy lights do not draw very many amps and the battery probably was taking a charge. Turn off all the accessories and charge your battery according to the instructions per charger and you should be good to go.

Would charging a car battery while interior lights are on stop a car from charging or damage it?

It depends. A filament lamp is essentially a resistor, and the interior light in your car is probably 5 watts. When driven by 12v it will consume about 0.42 amps, or when in a system that is more like 14v (i.e. when it's being charged) 0.36A. This applies for each bulb that is on.

In order for your car battery to charge, your battery charger needs to output enough current to supply what the lamp demands plus an excess that will be absorbed by the battery during the charging process. If your charger is a trickle type, and charges using some low current such as 0.8 amps, with a voltage of 14v (ish) your 5W bulb is consuming about 0.4 amps then you have around 0.4 amps for charging the battery..

Your car battery has a capacity rating, measured in amp-hours or minutes-to-reserve. It represents a theoretical ability to supply some amount of amps for some amount of time. Minutes-to-reserve describes the amount of time it takes to deplete a battery at a certain current. Amp hours describes the amount of time an amount of amps can be delivered. Mathematically the two are typically related by dividing Ah by 0.42 to get reserve minutes, or multiplying reserve minutes by 0.42 to get Ah.

An 80Ah battery (quite a high capacity) could (theoretically, if it was possible to usefully get all the charge out of it without losing energy to wasted heating) supply 80 amps for an hour, or 1 amp for 80 hours, or 2 amps for 40 hours, or... You get the idea. In a real world context a higher supply current generates heat and wastes energy so you wouldn't get 80 amps for an hour but for sake of simplicity let's assume perfect linearity

So if your battery was diminished to 50% of capacity, and you wanted to charge it to full then it's down 40 Ah. You need to supply 1 amp for 40 hours, or 40 amps for 1 hour, or... You get the idea.

Imagine that you were using a trickle charger of 0.8A and no light is on - it'd take 50 hours (2 days straight) to recharge your 50% depleted 80Ah battery. If the bulb is using half the current from the charger, the battery takes twice as long to charge

If you left two lights on you're probably in deficit - your trickle charger can't supply enough to run the bulbs let alone charge the battery

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Will it stop charging?

We can't really answer, because you didn't say what the output current rating of your charger is, or how many bulbs of what wattages were left on. You can though; you can read labels or measure the amount of current that is leaving the battery when your lights are on - the most exact science will be to disconnect one of the battery leads an put a multimeter in amps mode between the battery and the lead so you can measure how much current is flowing into or out of the battery. If it's flowing in, the battery is charging. If out, then the charger can't run the bulbs so the battery is helping, and suffering in the process.

Will it damage it?

It depends, but the answer is either "no" or "it'll actually help"

If your car battery is at 5.5v then it's already well on the way to being ruined. Standard lead acid cells lose ability to hold charge if they're depleted lower than about 80%. Batteries intended for leisure use (golf carts, mobility scooters etc) or marine use are designed differently and can withstand being depleted lower, but no lead acid cell can withstand deep discharge even if immediately recharged.

Your battery will never recover its former capacity. Whether this ultimately makes a difference probably largely depends on the environment. If it was in great condition before this, and you live in a region that is persistently warm year round, you routinely drive for long times between engine-off stops and there are no earth faults or other electrical maladies with the car then you might be able to recharge it and carry on with the diminished capacity. If you live in an area that frequently sees sub zero temperatures, or there is some fault that is a permanent parasitic drain, or you do a lot of stop start driving, you may well need to replace the battery because it will reach a point where it is no longer able to start the car

The best chance the battery has of recovering as best it can is to be recharged slowly. Fast recharging generates heat inside the battery, and excessive temperatures cause warping of the internals and internal shorts which further diminishes capacity. To this end, if you have a high power charger and no option to get a lower power (trickle) charger, applying some electrical load (leaving lights on) can help the battery survive an aggressive charging regimen because it isn't absorbing all of the output of the charger

I'd charge your battery on a sub-1-amp charger, and see how it recovers. If it does and is able to start the car, great. You might find that the combination of driving style and environment means you have to periodically recharge your battery overnight, and if e.g. is gets really cold you might have to replace it. If having a working car is vital, head this problem off before it begins and replace the battery; 5.5v is really low