It turns out that you can enable debug menu on Google Messages by entering *xyzzy* to the search field. It will add two additional items to the settings menu and a bunch of options to the dropdown menu.
The code phrase could be known to some people as a cheat code from the Road Rush game that was popular in the early 90s.
This menu existed in the app for a long time but recently it was highlighted by Reddit users as a way to fix possible issues with RCS. RCS is a Reach Communication Standard that is here to replace old SMS/MMS. This standard allows Google Messages app to send larger files, stickers, have group chats, and more.