Meus is the nominative case of the adjective meus, -a, -um (my) and matches with the nominative case subject discipulus.
Mihi is the dative case of the pronoun ego (I, me) and would thus be used for an indirect object (as in "He gives the gift to me"). It also gets used in phrases like "Canis mihi est" that get translated into English as "I have a dog" but more literally mean something like "There is a dog to me."
Yes, dative or possessor or of possession is very common. Est mihi pecunia, "I have money." Liber est puero, "the boy has a book." I'm guessing DL Latin will get to it later. In this sentence, what will you do with puer if you use that construction? You could elongate the sentence with Mihi est discipulus qui puer est, which might indicate that you have a slave that you are training to be an amanuensis for someone elite.