I started browsing the Wikipedia for Apartment, and came across this

One dwelling with two storeys

The vast majority of apartments are on one level, hence "flat". Some, however, have two storeys joined internally by stairs, just like many houses. One term for this is "maisonette", as above. Some housing in the United Kingdom, both public and private, was designed as scissor section flats. On a grander level, penthouses may have more than one storey to emphasise the idea of space and luxury. Two-storey units in new construction are sometimes referred to as "townhouses" in some countries (though not usually in Britain).

and

Maisonette ... has no strict definition, but the OED suggests "a part of a residential building which is occupied separately, usually on more than one floor and having its own outside entrance." It differs from a flat in having usually more than one floor

So it seems that flat implies one floor, and most exceptions have an alternative name like "penthouse"