I would argue that if you are able to make an attack from the echo knights position, your range is effectively what either you or your echo could reach. As you could make a melee attack from the echo, a grapple is just a "Special" melee attack. So it would be a standard roll from the echo's position. "When you take the Attack action on your turn, any attack you make with that action can originate from your space or the echo’s space. You make this choice for each attack." Based on the way the echo knight is written, the characters start used for attacks, (it even uses the characters saving throws after all) I would say that it would follow standard grappling procedure. The enemy creature-being-what ever was in range for an attack, because it was in range of the echo. If a grapple is an attack, and you can attack from the position of the echo. Then you can grapple from the echo.

Echo Knight is sloppy and is going to require some work from your DM

At this point, we can't avoid saying it. Usually, WotC writes very tightly and with great purpose. Echo Knight, however, has a lot of problems. If you want to play this class, make sure your DM is up for making a lot of rulings.

It seems clear from the language that the intention is for your reach to encompass the space around your echo, but the language is so imprecise and vague. The rules for reach are just kind of hinted at.

RAW reading - the grappled stays grappled

I'll try make a strict raw reading of the rules for you, but like I said before, this is going to be up to your DM.

The echo can be used to grapple: from Manifest Echo: "When you take the Attack action on your turn, any attack you make with that action can originate from your space or the echo's space.", from Grapple: "When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple." So for starters, you can use the echo to grapple.

The next problem is the Grappled condition, as you note: "The condition also ends if an effect removes the grappled creature from the reach of the grappler or grappling effect" notice the use of the word "removes"?

Remove means: "take (something) away or off from the position occupied." or "abolish or get rid of." The question is, has the grappled creature been removed from the reach of the grappler or grappling effect? From my reading, no. The grappled creature was never within your reach, therefore it could not be "removed" from reach. Alternately, if Manifest Echo is a vehicle for a "grappling effect", then the grappled creature has remained within reach of the grappling effect.

The grapple ends immediately

Given the fact that a grapple is an attack, which the rules permit the echo to do

No. Echos cannot attack. You can perform an attack as if you were in the echos space. So you could certainly try to grapple the creature, but whether you succeed or fail you could not sustain the grapple for longer than the instant of the attack if you're not already within grappling range of the creature you're trying to grapple.

The problem is that the echo knight is never moved away by an effect because it was never within reach to begin with.

This doesn't matter even if we're being extremely pedantic, because the actual text says

The condition also ends if an effect removes the grappled creature from the reach of the grappler

Nothing in here says that either the grappled creature or the grappler have to move, this says that the grapple ends when the creature is no longer in reach for the grapple - of course much more commonly this would be the case when either is forcefully moved away from the other, but in this case it is "removed from your reach" because your reach itself has changed. A simular situation would occur for instance if a Enlarge'd (from medium to large) grappled creatures on either side of it and the Enlarge ended - clearly nobody is moving in this circumstance, but the creature no longer has the reach to grapple both of them so the grapple ends on at least one of the creatures.

Or is it? The rules for Manifest echo effectively grant the knight reach from that space because the knight can make attacks of opportunity from the echo's space.

No, nothing in the wording of the feature indicates that it actually extends your reach outside of attacks. You cannot even really make attack of opportunities as if you were in the echos space, as it is worded it doesn't work with the Sentinel feat (except for the first bullet point) for instance.