Apart from giving users that are part of a family a bonus to their streaks, it really wouldn't increase the interactivity between users at all since there would be no special interactions. It doesn't really make sense to implement something like this.
Although, taking your idea and changing it a little would have some potential. However;
They wouldn't be called a family, but a group or a guild (makes more sense, since family would kind of imply relatives) and from here you can pretty much do whatever other guilds in every other game do
As players would be parts of guilds, they'd be able to fight other guilds, which would add a whole new level of interaction to the bot, but also substantially change the way the bot works (think about it, rn the only thing you can do with pets is walk them and train them to no end) and would also require a lot of coding and changing. It would, however, increase interaction between users substantially. But then again, even pvp of any sort would do the same thing
Your idea of a family just doesn't make a lot of sense in any regard, as it really doesn't bring anything aside from belonging to a group. You can be more creative than that ;) I believe in you!
Kind regards,