Is it possible to limit name and photo sharing to certain contacts on the Messages app?

I have recently started sharing my name and picture with some of my contacts. I was wondering if in the settings somewhere we can see which contacts we have given access to see our name and photo.

If later I decide to remove this access for one of my contacts, where can I do that without turning this feature off completely for all of my contacts?

[Re-Titled by Moderator]

Original Title: Photo and Name Sharing with Contacts