If you use a Bluetooth speaker in your car, yes it should still work with Car Thing. I've tested this before. If you don't have a Bluetooth Car Stereo or Bluetooth speaker connected to your phone, then it will playback through your phone's speakers by default. Once you connect a Bluetooth Car Stereo or Bluetooth speaker, then the sound will come out of that.
On the setup page here, instead of connecting your phone to your Car's speakers, you would connect your phone to a Bluetooth speaker: