If I start Drafts and turn on Enable menu bar in settings, Drafts appears on the menu bar and functions properly. If I return to drafts.app and quit it, the menu bar item disappears. If I start Drafts from the menu bar and quit it from the menu bar, it deletes the Drafts item from the menu bar, as it should. (If I repeat the test with Day One.app, the menu bar stays even if I close the app. As expected, quitting Day One from the menu bar removes the item.) macOS Version 15.0 Beta (24A5320a)
I think the Drafts menu bar entry is the Drafts app rather than a separate helper app. You close Drafts, you close that. Apps often use helper apps whereby you close the main app, but the mini helper app stays resident.