My wife has full custody of our son. He is my step son but I consider him my own flesh and blood and love him like a real son. I have been in his life since he was 6 months old. I want to know if the father will still have to give up his rights to "his" son if my wife has full custody?

Yes of course, full custody does not take away the fathers rights to visits if he wished to file or them. ( unless he has them) So yes the bio father will have to be asked and permission given