The extension of I, we, you, they don't to the third person singular he, she, it is a case of paradigm leveling, where distinctions within one morphological pattern are leveled out in favor of a single one. Leveling is a force operating in virtually every language. It's why strive has become a weak verb for many English speakers and why Slovak (though not Czech) no longer bothers with a productive vocative case.
For many speakers on both sides of the Atlantic, the leveling of doesn't to don't only extends to the negative, yielding this question in a short story set in a pub somewhere in England:
He don't do that, does he?
Third person singular don't would simply be some charming but often deprecated dialect usage were it not for one particular cultural phenomen: speakers who regularly employ this leveled form have produced an inordinate amount, even whole genres, of popular music.
A prominent feature of both African American Vernacular English(AAVE) and that of the Mountain/Inland South, he, she, it don't was destined to appear in genres such as rap, hiphop, rhythm and blues, Motown, gospel, spirituals, minstrel, ragtime, and some jazz as well as Country-Western, bluegrass, white gospel, and folk songs either sung in these dialect areas or meant to sound like them.
With a few exceptions, singers in these genres will often imitate the associated dialect features even if their own speech is nothing like it. Singer Tom Jones is as Welsh as leeks and Charlotte Church, but because of the genres he sings, he adopts many features of AAVE, as when he covers "You Can Leave Your Hat On." Willie Nelson couldn't be from anywhere but Texas, but one hit wonder Jeannie C. Riley, from Anson, Texas, was aiming for Nashville in "Harper Valley P.T.A.".
Except in the company of language purists of little imagination, dialect or non-standard forms in song lyrics suggest familiarity, intimacy, and a certain rawness of emotion. Imagine Pink Floyd singing "We don't need any education" or Elvis lamenting "I'm nothing but a dog."
Falling well beyond the reach of such generosity toward dialect forms is grammar tortured for the mere sake of a rhyme. The Christmas song "I Wonder as I Wander," covered by many artists, even Streisand, begins with the verse:
I wonder as I wander out under the sky,
How Jesus the Savior did come for to die,
For poor on'ry people like you and like I.
wonder as I wander out under the sky.
Come for to die and on'ry are vaguely Appalachian, but then comes the cringeworthy like I. At best a hypercorrection one would expect from Hyacinth Bucket in "Keeping Up Appearances," it exists solely for the rhyme. This isn't just bad grammar; it's bad poetry.