What happens when you hatch an egg with a full pokemon box?

No. You lost an incubator and gained a mon. Net zero. And the egg counts already

You have a wrong assumption, the eggs take a Pokemon storage as well as item storage , so the unhatched monsters is already in consideration
Also you can check in youe journal what was hatched ( if soon enough )
Or you can arrangw your Pokemon chronical wise and see the monsters with 2500 dust is mostly hatched

I think that or an egg already counts as a pokemon or it works like when you bag is full and you lvl up, you just stay with like 460 of 450 items.

Yeah what they already said, eggs count against your storage space. That's why you'll never get an egg from a stop if your storage is full even if you have egg slots available.

You can't get an egg if your Pokemon storage is full. Your Pokemon storage number increases by 1 when you get the egg, rather than when it is hatched.

If the Pokemon were recently hatched then check your journal and/or sort your Pokemon by recent, and if that isn't enough, check the IVs of your recent Pokemon. The ones from eggs will have 10/10/10 IVs minimum and be level 20.

It's still possible that you could have caught Pokemon with good IVs and mistake them for egg hatches but that's the best you can do if the egg hatch isn't in the journal anymore.

Just sort your pokemon on recent. First 2 are your hatches.