Were knights expected to go into the service of whoever trained them?

Ive heard over the years that once a knight's father sends him over as a page/squire to another noble, he's essentially signing him into his service.

How are knightly orders a thing then? If you're put into the service of someone before becoming a knight, how come there are so many groups formed mainly of knights? We're you able to just leave your lord's service?

Also, what if the noble who trained you didn't have a small castle or fortress to make you burgrave of? Would the only nobles who take in squires be influential enough to have such area, or could any nobility take you in so long as they were at least slightly more influential than your father?

I would like to note that most of my knowledge comes from bohemia which I understand functioned quite differently than the rest of Europe.