You are right. The correct form is 'It doesn't matter'. This is a straightforward negative statement in the present simple: I don't matter = I'm not important You don't matter = You're not important It doesn't matter = It's not important 'It don't matter' is what we call a non-standard form. It doesn't follow the rules of 'correct' grammar, but it is commonly used in some dialects and by some less educated people. Why do we come across non-standard grammar so much in popular songs? One reason is that it often sounds better. 'It don't' may fit better with the music than 'It doesn't'. The same goes for 'ain't', and other non-standard forms. Another reason is that the songwriter may want the message of the song to sound cool and rebellious, angry or emotional or more 'immediate'. They may want to appeal more to younger people who want to break away from conventions. There is more 'street appeal' if a song uses non-standard grammar. 'No matter' is a neat little set phrase equivalent to 'It doesn't matter'. It's quite old-fashioned, though. You very rarely hear this nowadays.

Don't matter: Non-standard use predominantly to sound cool by youngsters. Eg. - It don't matter if the explanation is not good enough. I'll validate it to be sure. Doesn't matter: Grammatically correct. Eg. - It doesn't matter if the explanation is not good enough. I'll validate it to be sure. No matter: Used in a sentence to introduce a conditional clause followed by the main clause. Eg.: No matter how difficult this explanation may sound, I'm gonna go to the depth of it.