Is this herb actually called Lemon Balm in the US?

I cannot see the leaf texture, but I can help you try to identify what you have. I grow both at the moment, and both are members of the mint family, which can make it tricky. Lemon Mint has smooth leaves with more sharply serrated edges. Lemon balm has a rougher, bumpier leave, with more rounded serrated edges. Only the lemon mint contains menthol, so the lemony smell is less intense with lemon mint than with lemon balm. If you are growing it, lemon balm is a bit friendly with neighbor plants. It doesn't spread via runners, as lemon mint will do. It self seeds.

There is an herb called lemon balm in the U.S. It has elongated, tapered leaves that are slightly fuzzy. It does look similar to mint but has a definite lemon smell. I would not be surprised it it is in the mint family.