I was on Catalina last week, and recently upgraded to Monterey. When doing so, the Messages App lost it's ability to simply "Close conversation", which used to preserve the entire history of the conversation; when I would start a new chat with a Closed conversation contact, all the history would re-appear for my enjoyment.
As of Monterey Messages, there is only the option to Delete Conversation - and it warns that it will be deleted, of course. This behaves just like the iOS version of Messages, of course. However, on the macOS, I always kept my history. Now this option seems gone. FYI, I would formerly "Close" convos just to keep my list of active convos clean and tidy. Now, I guess I have to keep every conversation active for all time to preserve the life-long history which, in some business situations, is required.
I am responding to this topic because the OP wanted to know how to Archive, and felt the need to point out a loss of functionality in the new(er) iOS-like macOS Messenger app.