Yes, the carry limit applies per person, so you could, in theory, have an entire family wherein every single member has six pokémon each. It’s just not ideal for practical reasons, which is why a lot of families will only keep one to three pokémon on hand per family member. After all, if you’re a family of five, it gets rather difficult to feed, clean, and accommodate yourselves and thirty pokémon unless you happen to own a lot of land.
There are, of course, exceptions to the six-pokémon limit. Some buildings or communities (particularly those in urban areas) may prevent trainers from carrying a full six if they’re living with other trainers. Alternatively, they may force trainers to keep their pokémon in their poké balls at all times. And of course, if you have a service pokémon, it never counts towards the carry limit, so you could even theoretically have more pokémon in a household than just a multiple of six.
It depends on a number of factors, in other words.