The new "Recently Played" section is rolling out to the mobile app in the home tab. Here you will be able to see up to 3 months of listening history. Look for the clock icon in the top right corner of the home tab.
Travel back in time: Rediscover lost gems in your listening history with a new “Recently played” destination, where users can jump back in time and browse up to three months’ worth of listening history. Premium and Free users globally will be able to browse recently played individual tracks and episodes in addition to the playlists, albums, and shows they were played from.
You can read more info about "recently played" and other updates to the home tab in the news article:
New Features on Spotify’s Home Hub Make Navigation and Discovery Even Simpler
I just had the desktop app open while I was playing music in the mobile app and the songs played in the mobile app showed up in the history section of the desktop app for me. Not sure if the desktop app has to be open for this to happen but this seems to be an improvement over the previous behavior.
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