Can my daughter's father relinquish his parental rights just because I want to change our daughter's last name from his to mine? Do I have a say in this?

NO! There is a sticky note at the top of this forum that discusses terminating parental rights. A parent cannot "relinquish" their rights. Only a court can issue a TPR. And courts are very reluctant to do so.

Nor can you arbitrarily change her name. Her legal name is what's on her birth certificate. The only way to change that is via a court order. And the father would have to agree to it.

I know that I have to do it through the courts. I have the paperwork. I notified her father I was going to do this and he said he would relinquish his rights. I was just wandering if he could do this?

No, he can't do it. But he can block you from making the change.

Even if he has nothing to do with her? It's just an "ownership" thing for him. He is a sociopath come to find out.

He is the legal father and if he objects to the name change its probable a court will not grant it.