The microphone icon appears when the song has lyrics. When an instrumental song or song that has no lyrics submitted to Musixmatch starts playing, the microphone icon disappears.
Musixmatch is a service where lyrics are submitted, and that's where Spotify gets the lyrics from.
If you have no microphone icon no matter what song plays (try some very mainstream songs, they 100% have lyrics submitted to Musixmatch), then you may have an outdated Spotify app.
For reference, I have this version of Spotify from Snap, and it has functional lyrics feature:
Try and see if there's an update for Spotify available on your OS. : )
Let me know how it goes!
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Note: I'm not a Spotify employee.