Interesting.. I would never think Condo described the financial relation you had with the house but not some intrinsic quality of the house itself.

Technically it doesn’t (as it means condominium which in theory could be single family homes) but practically it almost always means an “apartment you own” though sometimes “half a duplex” or similar proportions of a quad or six plex style.

But in the US it usually implies neighbors above and below and on the sides.

I'm more curious now: what sort of apartment does not have neighbours nearby?

It’s the above and below; most people in the USA will think “condominium tower” when you say condo, but some (rarer) are much more like duplexes or single family homes.