Iphone ownership transfer

There is no ownership registry for the device. You're presumably logged into your iCloud account. As far as anyone knows, it is yours. You could trade it in toward a new phone if you wanted to do so.

For the cell plan, you can be added to the plan.

This other person is intentionally playing a very cruel game, and could take the phone back. "Gift" is often adjudicated in courts after a complaint is filed, and if this other person did pay for the device they has an official receipt (if you think they have not saved the receipt, think again!) they can get officials to force you to turn it over.

Buy your own phone, get your own plan with another carrier and try to transfer the existing number ... then return this "gift". If you cannot transfer the number, accept that a new number is a required part of getting at least this much independence. If you need personal physical privacy (the other person cannot physically hold and therefore alter the new phone by adding malicious software) then arrange the new living site first.

Yes, this is not "just a technical" answer, but this if people feel it should be deleted for that reason I can accept the enforcement of the forum rules.

Phone service, sure. Anyone on the account could make changes.

physical phone, it’s a gift. They cannot simply take it back (at least not in most jurisdictions).