I've never heard of such a thing, and have been driving for many years. even a quick google search didn't find anything other than questions as to why there aren't special licenses.
can you point to an example state that has these requirements?
in my part of the country (pacific nw), there are often social media posts asking for someone to loan them a manual transmission vehicle so they can get "used to a clutch".
My definition of here is Norway - if you cannot drive a manual transmission, your license is has a restriction. You can get ticketed for driving a manual transmission instead of an automatic.
I was lucky and learned to drive one in the US well before I moved here, but I imagine one would need to either go to a driving instruction school, get a special permit to learn to drive one with a qualified driver, or minimally take a driving test after having illegally learned to drive one before removing it.
Edit: I can't help but to comment that I had one better. I had a 1990's hatchback. The rear hatch didn't lock, but you had to do the right motions to open it lest it act locked. It was an automatic, but didn't have much "oomph". Finally, the handles to roll up the windows didn't stay on the doors. I stored them on the dashboard. (they would still roll up/down the windows, but would not stay).