Apple Support has shared a new video detailing how to share your name and photo within the Messages app on iPhone, iPad and iPod touch:
In iOS 13 and iPadOS, you can share your name and photo—even your Memoji—with the people you chat with in Messages. You can share your name and photo with all of your contacts, or pick and choose who should see it.
You may have noticed many of your contacts now showing a different name and look within Messages, thanks to iOS 13 and the ability to create a Memoji. The following video details how you can do the same or use a different photo for your contacts to see: