Since 2006 I've engaged in a bit of navel-gazing and tracked my music listening using last.fm. By running an application on your computer or phone, or linking a streaming service to last.fm (e.g. Tidal and Spotify support this), the service keeps track of what songs you listen to. The act of recording a song listen is called a "scrobble." However, it's always kind of bugged me that you can't scrobble in all situations, like when listening to an internet radio station or terrestrial radio.
Recently I discovered the Finale app that does exactly this: it not only allows you to manually enter songs, it includes Shazam-like ability to listen to songs and identify them for you, and if you like, then scrobble the song. It has a "continuously listen" mode which despite the name means that once a minute it wakes up and listens for any new song you're listening to and scrobbles for you.
Somewhat related, I would like to recommend The Colorado Sound, a music radio station operated by a local NPR affiliate. It's as commercial-free as NPR is (limited to "supported by" messages) and has a very wide and eclectic selection of music. Using the Finale app while I listen to their internet stream allows me to keep track of what they're playing, and if I hear something I like, I can revisit that song/artist. You should check it out!