It is news to me (which would not be surprising in the current chaos of things) that similar arrangements are in place for AMEs because there are problems of privacy.
The only problem area I am aware of is the German LBA and SOLIs as stated elsewhere. That’s not to say there aren’t any other issues, but in general this practice has been accepted for years now.
We’ve had U.K. licence holders renewing Class 1 medicals in Spain, Irish licence holders renewing medicals in the U.K. (myself included) and vice-versa. I’ve seen Polish medicals accepted by Spain - the list goes on.