Family is extremely important to a lot of people. It is also an institution we have had as long as we have existed.
If we wanted to, how hard would it be to create another institution with similar virtues? (Not to replace families in general, but to have more of that value, or a backup for when family fails, as it does for many people.)
People often consider close friends to to be their ‘chosen family’, but I think the inability to choose is relevant to some of the virtue of real families: you are stuck with each other almost no matter what, and you are forced to be on good
A rough method I could imagine: take a large set of people and use AI to sort them by some measure of personality similarity and physical proximity into groups of 4-10, across different ages. Give each group substantial incentive to interact and be on good terms with each other, for instance free use of local amenities if they show up together. (I’m imagining this is brought about by a national government, but alternately a large group of people could decide to do this with each other.)
This plan is probably missing something like the meaning and loyalty-to-ones-blood aspects of family. Though I could imagine the right personality assessment system might change that, if people felt they were being put in touch with essentially their platonic soul-mates.
You need to read the Terra Ignota series, it’s a science fiction and their whole society is based around your concept
Robber's Cave experiment suggests maybe not much more than creating the groups would be necessary