JOe, thanks How new is this pinning feature?
It is in iOS 6 so at least ten months old. (I am not a good judge of that because I run beta software and stay four or five months ahead of the general release curve.)
Can the other person pin it, or am I'm pinning inadvertently?
You are doing it inadvertently. Almost certainly by holding a click too long.
Also, I don't see a way to right or control click a conversation, on the iphone. I do see a way if I'm on desktop in messenger and have mouse.'
Even then I don't see a way to unpin it.
You may not like the answer, as it has caused you so much aggravation already but a LONG press
on the conversation icon — not the text — will bringup a context menu with options to pin or unpin, hide alerts, or delete. NOTE:
A long press (a.k.a. "long touch") is the touch interface (iOS, iPadOS) equivalent of control+click on the Mac keyboard, a right click on the mouse, or a two finger tap or click on a trackpad. Pinning or unpinning a conversation on one device pins or unpins the conversation on all devices on the same Apple ID account.