It is pretty simple and common idea to have a button on the music apps to show lyrics, coz you feel it is very much needed sometimes.
I guess it is because of copyright/authorisation issue, most apps don’t have it. One has to install apps like Musicxmatch to get live lyrics of the song you are playing.
However, Google shows lyrics of most songs when you search for, with the copyright info within.
Google should tweak a bit to introduce an option on the Google Play Music app to show lyrics.
The current line of the lyrics being played can be highlighted like the apps like Musixmatch do.