The thing is, he wants to distance himself from his father and make it on his own. He goes by the surname "Johnson" to avoid anyone knowing who he is, and whenever someone mentions it, he tells them "I don't know him. Ain't got nothing to do with me." When he moves to Philly in order to seek out Rocky as his trainer, he insists "Every punch I've ever thrown is my own," and that Apollo has nothing whatsoever to do with his ambition or success. And when he finally gets that big fight against the world champion, he got there on his own, not as a shadow of his famous father.
Of course, none of that makes any sense. Are you kidding? It has everything to do with his father. From the very beginning, he's only able to quit his job, move across the country, and train full-time without taking on a side job is because of Apollo's ridiculous wealth.
This was not a man of austerity.
The only way he's able to convince a long-retired Rocky to be his trainer is, again, because of his dad. Anyone else who mentions Apollo to his face is likely to get punched out, but Adonis doesn't approach Rocky under his pseudonym -- he name-drops his dad within two minutes.
Then he's offered a showcase fight on HBO despite being woefully inexperienced. Why? Because they can promote him as Apollo Creed's son. So does he reject the offer to pad his record and organically build up his own following so he can eventually earn the shot on his own? Nope, he changes his name back and goes with it.