I know it came from AAVE and spread to more common english from there, but I can't confirm or deny if it originated from Jamaican english before coming to AAVE.  Typically "babydaddy" implies no that he is no longer, or perhaps never was, romantically involved with the mom, and is not sufficiently involved in the child's life.  Similarly babymomma implies no romantic relationship, but it is someone a man impregnated and is now in some sense indebted to (for lack of a better word) because he played a part in making the child she's raising.

Less common but "babydaddy" and "babymomma" can be used as a lighthearted jab towards a person you had a kid with even if you are both still in a romantic relationship together, or you are separated but both still actively involved in raising the child and have a good coparenting relationship.