Classic Edition 3.0.226.993 (GoG)
I also have an issue with multiplayer that sounds identical. We (both using GoG) were last able to successfully connect and play on October 11th. After that the functionality from both the Invite Players button on the main menu as well as the invite buttons on the bottom of the Connectivity Menu simply do not work. The buttons animate when clicked but no longer do anything.
I think it is related to the save file in some way. My computer had the issue first so we tested by transferring the save file to the second computer. It was able to create a new game and invite players as normal before loading the save file. As soon as the save file was loaded the second computer experienced the exact issue with broken invite system as well. Unfortunately, deleting the save file doesn't help and the issue persists.
We've run through the full troubleshooting list including verification of game files, new profiles, etc. I just did a full delete and reinstall, made sure all saves were deleted, created a new profile, started a new multiplayer lobby, and experienced the same issue.