It seems that play counts are totally disfunctional in the latest versions of iTunes (versions 12.5.3 currently). I know that there have been sync problems all the time, but since I joined Apple Music it won't even increase the count for songs played from the local library in the local iTunes. Most of the songs in my library are stuck at zero plays although I listen to them often.

Everything else in Apple Music seems to be working all right. In particular, the "For You" Section does show my latest plays under "Recent plays", just the numbers in the library remain zero forever. I wonder if it is relevant, or if maybe Apple Music keeps playcounts separate from the local ones.