You are 100% right that "it doesn't sound the same" is grammatically correct while "it don't sound the same" is grammatically wrong.

It is fairly common for songs lyrics to have grammar mistakes.  Sometimes singers do it because the sound of the "less correct" word or the number of its syllables just matches the music better.  Sometimes they do it because they want the song to sound more casual (not like a carefully written essay for school!).  Sometimes they want to give the feeling that the song is being sung by someone from a certain region or cultural group.

My Spanish-speaking friend is a big fan of the country singer Shania Twain.  I told him that is fine, but to be careful not to model his English after her English because she uses lyrics like "I ain't no quitter" (the correct English would be "I'm not a quitter") and "That don't impress me much" (the correct English would be "That doesn't impress me much").  He was really puzzled and said, "But she seems like she is a smart lady.  Why doesn't she know the right way to say it?"  Ha, I'm sure she knows very well how to speak properly, but she intentionally uses very casual words and poor grammar in order to give her songs a certain style.