How to set up Name and Photo Sharing in Messages app on iPhone 16 Pro Max?

My wife asked for help in sharing her photo and name in Messages. She uses an iPhone 13 Pro Max). I went to her Settings >> Apps >> Messages >> Share Name and Photo and selected the photo she had chosen to share.

I clicked Done and went to her Messages app, where I sent a text (iMessage) to myself. (I have an iPhone 16 Pro Max. And yes, both phones are fully updated.) I got the text message, but nothing appears to have changed. I see the same photo of her at the top of the screen—it’s the same photo that appears with her contact sheet in my Contacts app.

My question then is, exactly how does this sharing of her name and photo manifest itself to the recipient? I’ll assume that the recipient uses an iPhone, not some other device. Why don’t I see the photo that she chose to share?

[Re-Titled by Moderator]

Original Title: Sharing name and photo in Messages app