Apple Music's Recently Played page is meant to serve as a hub for viewing your watch history, but it's somewhat confusing. With iOS 13.2, the app now has a history page that lets you view all songs you've heard with just a few taps and swipes.
To access the Apple Music history, open the app and tap the Now Playing bar (the current song) at the bottom of the screen. Then tap the three-line menu icon in the lower-right corner and swipe down to scroll up over Next Up. At the top left you will see the word "History" as well as a list of all the songs you have heard in the past.