The caller ID is provided by your carrier, it does not come from your phone. This means that your carrier has his name on your phone number. Call the carrier to get this fixed, or you may be able to fix it in their web portal.
As a separate issue, you should not have 1 Apple ID; you should each have your own. Apple IDs are personal; they shouldn’t be shared any more than you would share a toothbrush. If you share an Apple ID with someone else many bad things will happen. Each user should have their own Apple ID, then you can create Family Sharing→What is Family Sharing? - Apple Support. That way you can share what you want to share and not share what you don’t want to share.
It is OK to share an Apple ID among devices used by one person, such as an iPhone, an iPad and a Mac; in fact, it is a very useful thing to do. But not between devices used by different people.
Here is Apple’s FAQ: https://appleid.apple.com/faq/#!&page=faq. See the 4th bullet item.