icon to the right of photos on iMessage iPhone - iOS15

Are you referring to the icon next to a photo that is sent to you in the Messages app? The icon looks like a box with a down arrow in it? If you click on it then it gives you the option to SAVE PHOTO to your Photos app.

Axel F.

There is no setting to turn it OFF. It is ON by default and will remain ON.

Axel F.