I've always kept it simple. Except if you took advantages or disadvantages reflecting this in background, your name is the one of your parents, and you serve their lords as your ancestors probably did before.
for players, as long as they pay for it and role-play the drawbacks, I'm OK with multiple and different schools.
for NPC it's a bit particular. In my Rokugan 4e, Not everyone goes to schools. The mass of samurai still study and train at dojo with Sensei, their abilities comes from skill ranks and non school restricted Kata, but schools are attended only by the worthy or the wealthy.
I also think family trait represent your genetics, while school traits would be your training.
so yes, folks unschooled would only get a family trait.
it sets the PCs apart and make named npcs with techniques more impactful encounters and technique and schools less mundane.