He explicitly mentions his mother in the very start of the song ("Well, my daddy left home when I was three, and he didn't leave much to ma and me"). I think one can assume in whatever region he lives she wouldn't have any say on what to call him since "the man" had said it would be Sue and that was final! I guess she could have done that out of spite for her husband, but... just didn't.
It's either Rule of Funny or the mother knew the reason why the father named him Sue and thought it was a good idea.
How else would we have the line, "My name is Sue. How do you do? NOW YOU'RE GONNA DIE!"
"One Piece At A Time": After the narrator devises his plan to acquire a Cadillac from the auto plant where he works by smuggling the parts out sequentially in his lunch box, he says
The first day I got me a fuel pump
And the next day I got me an engine and a trunk
And then I got me a transmission and all the chrome
But when he finally has all the parts and begins to assemble it, he says:
The transmission was a '53
But the motor turned out to be a '73
How can the transmission be twenty years older than the motor when he explicitly says earlier that he got the motor first?
Maybe he stole a old leftover transmission from a storage area?