If you wanted to sync all your music manually, you could just copy it from one computer to another every time you got a new song. Of course, the point here is to avoid that kind of tedium and find an easier way. If you want to avoid any special software or services, however, you can use watch folders to automatically add music to your libraries with little effort.
Pretty much every music player has this feature, but it can work in a couple of ways. The best way is if you can specify a location where the player watches for new music to add. This way you can set that location and every source can draw from it (over the network or locally) presuming the original file is copied rather than moved.
The other way watch folders can work is if a specific folder is designated by the music player. Any music added to that specific folder will import. (This is how iTunes does it.) That makes creating a syncing solution a bit harder because you can't just download songs and expect them to be added automatically. You can, however, use an app like Belvedere or Hazel to copy newly downloaded music files to multiple watch folders for you to create a sync-like experience.