I just read Rav Eliezer Melamed’s answer to a question saying it is forbidden to copy a CD for personal use if it doesn’t belong to you. If a person has a CD that belongs to them, are they allowed to copy it, as they own it, and then give it to someone as a present because technically he owns it now?

It's the same as if he does it, since you have not the right to duplicate it for others.