There are a great many songs available in the world. There are so many that I would be extremely surprised to learn that there isn't at least one—probably several, in different genres—about each of the 24 hours of the day, and each of the seven days of the week. There might even be ones for each quarter-hour of the day.
Therefore, this is a clock would be implemented as a phone or computer app that connects to a music streaming service, because you're unlikely to have all of the necessary songs in an offline music library. Every hour, on the hour, it plays a song about that time of day. For example, at 05:00, it might play "5 in the Morning" by Charli XCX, if that matches the music genre preferences you've set. It can also avoid playing music while you're sleeping, either with a simple "don't play between these times" setting or by connecting to your sleep-tracking app to see whether you're actually in bed. (I think there used to be some kind of community-standard sleep app integration protocol on Android, but I doubt it's still in use.)
Every morning, when you wake up, it can play a song about that day of the week (e.g. "Friday" by Rebecca Black), or even that specific month and day (e.g. "9th of october" by Tove Lo), if there are songs available for all of those, though it might have to deviate more from your genre preferences for that. It might also have to deviate more if you want to hear a song every quarter-hour.
It can also do one of the following (user-choosable) if some audio is already playing when the hour changes: skip playing a song about the current time; delay playing the song about the current time until the other media is over; pause the other media and resume it once the song about the current time is done; insert the song about the current time into the queue of other media. This can also be set to depend on what the other media is: if you're listening to music, you might want it to insert the song about the current time after the currently-playing song; if you're watching a movie, you might want to not hear the song about the current time until the movie is over, if it's even still relevant by then.
N/A [2019-12-07]-- notexactly, Dec 07 2019 I have a horrible (gifted) cloc that plays Muzak versions of Beatle songs-- theircompetitor, Dec 07 2019 random, halfbakery