Sorry to invoke a cliche here but... Europe doesn't start and end in Portugal.
There are plenty of large-family apartments in some cities. Especially new developments usually have a mix of studios, one-bedrooms and beyond.
In terms of old buildings (i.e. older than ~100 years), many European cities have apartment buildings designed for wealthy families that would house their staff in the same building. The problem there is actually that these are too large to rent out to most families, not the opposite.