How to change my caller ID name on my iPhone?

What's displayed to others during a call is based on what they've saved in their contact list, not what you've specifically shared through Message settings.

When you utilize the Photo Sharing feature within the Message app, the recipient might have the ability to save that photo and associate it with your contact information. Unfortunately, this isn't something you can modify or control once it's stored on their end.

Nevertheless, you retain the power to discontinue sharing your photos through messages at any given moment. This allows you to manage the ongoing distribution of your images through that particular channel.

However you can stop sharing your Photos through messages anytime.

Please see the pic below which is self-explanatory.

Start Sharing:

Stop Sharing

When you share a photo or name with someone, the recipient has the choice to either update or disregard the information you've provided. It's important to note that this data isn't automatically stored on their device without their explicit action to save it. This gives them control over what they want to do with the shared content, ensuring their autonomy and privacy in managing the information they receive from you.

See the pic below