Yes, if you hold the button, you get continuous transmission. You can see that looking at the LED on the receiver, it's on continuously as long as you hold the key.
By changing the jumpers on the MP3 module, you can affect the behavior of the playback. The default shipping configuration is 010, but obviously the user can change the jumpers to select any of the options. The 001 would indeed allow a play & stop function by repeated presses of the keyfob key.