Flat = Apartment

Flat derives from a (non-English?) word for floor or level - hence a round tyre (tire) without air in it can be “flat”. Flats were single story houses stacked on top of each other.

Your comment is interestingly another example in itself of how we speak different languages.

In British English you generally can’t use “anymore” with a positive sentence because “anymore” means something was the case, but is now not.

“We don’t speak the same language anymore” - OK in BE

“We speak different languages anymore” - Not OK in BE

You’d have to instead say:

“We speak different languages these days” - OK in BE

I always interpreted it as an apartment in a tall building. Each apartment is a small one-level unit.