What would you call a separate second floor of a house (not an apartment, flat, attic, etc)?

Perhaps you can call it a (residential) unit.

ODO:

unit
NOUN

1.1 A self-contained section in a building or group of buildings.
‘one- and two-bedroom units’

‘The county council has applied for planning permission to convert the listed building into three residential units and once this has been granted will put it on to the open market.’

I think a literal translation of the Turkish term "aile apartımanı", ie. "family apartments" would fit the bill.

The units here are architecturally indistinguishable from regular apartments. The only difference is that they are mostly collectively owned by a family (traditionally registered to the family patriarch's name, but not any more after a few generations). Each apartment is occupied by a single nuclear family within the greater family tree.