There are no files (or anything) "saved" in Recents. It is a Search folder that looks primarily for the kMDItemLastUsedDate metadata attached to a file by Finder when you open the item from the Finder. A search folder finds items that match its criteria and displays them on one location. There is nothing saved into or moved into a search folder as it really isn't a folder. It is just all the items found by the search. The items in the search folder are, for the most part, the actual files that are stored somewhere on your drive. You can do anything with the items in a search folder that you would do with the real item, including deleting it.
If you open a file from the Finder, it should end up in Recents. If it is not, your Spotlight Index might not be complete.
Theoretically, an app could attach the metadata when it opens a file, but I don't know if there are any apps that do that.
Rebuild the Spotlight index on your Mac - Apple Support