There are multiple meanings of the word 'own'. It can be a verb, a pronoun, and a determiner, so it isn't surprising it could be used more than once in a statement. For example, it is fairly common to say "I own my own home", in which the first use of the word is a verb for ownership, and the second instance works as a determiner to indicate whose home it is (obviously, it is quite common to own a property that somebody else lives in, and to live in a property that you do not own).
Without the full lyrics of your song I can't be overly specific about the meaning but it's pretty clear your quotation is using the word firstly as a verb meaning to take ownership of, and then as a pronoun referring to what is yours. "Own", in the sense of ownership, has a common modern usage which means to gain control of something, particularly with reference to your life. My assumption, without further context, would be that "got to own your own" means you should take control of what is already yours - perhaps your life.