In your list of conversations in the Messages app, you probably have lots of people who have generic icons next to their names or numbers. You likely look like that to other people as well, but a new feature in iOS 13 lets you share your preferred name and avatar picture with other iMessage users (blue-bubble friends).
In Messages, first tap the ••• button and then Edit Name and Photo. Then, in the activity view that appears, tap Edit under your photo to select a photo or Animoji, and set your name as you want it to appear to others. Choose wether you want it to be automatically shared with people in your Contacts. Make sure Name and Photo Sharing is enabled before tapping Done. From now on, for any iMessage conversations, you’ll see a little banner at the top that asks if you want to share your name and photo. Do so and next time you send a message your recipient will get a prompt to replace whatever they’re seeing for you. (And if, as a recipient, you don’t want to accept the new photo, tap the X button at the right of the prompt.)