The reason we don't say we are hodling UTXOs is because it's kinda vague since we are hodling the coins in a wallet. It's a little bit similar to having a property and tying it to your company which you are legally seen as a trustee instead of an owner ( especially to avoid taxes) when you intend addressing yourself you would say you are the owner which is true although technically speaking you are a trustee.

They remain your coins because they haven't been spent and that's factual but the concept of UTXOs usually comes in when we go quite technical so you still have a valid point.