I think this is how it works - I haven't actually tested this but it seems to be the case.
The growth of the tam on V4 is based entirely on skill points - the randomness of taking care of the tam dissapears after the teen stage.
However, I think the randomness of the family you get corrolates to the randomness of the skill points. This could easily be disproven by someone with a little more V4 experience, but it seems logical that the highest number of Skill points in a particular catagory determines your family at the toddler age. E.g. You'd get the mametchi family if you got the most intelligence skill points out of the random games because the family is based on intelligence, Kuchi family if you get the most social from random points since they have the Social special character, so on, so forth. Basically, although the skill points from toddler games are random, the highest number of points in one area still determine your family.
The randomness of the skill point distribution at the toddler age may be what causes the family gotten to be random.
Then, after you've grown into a particular family, you have complete control over which character you get by playing certain games.
Universal characters are probably gotten by having a low total skill point sum at the time of changing.
Just a theory, but I think that the skill point feature is the driving force.