You can do this, after app updates. I had a 2013 Moto G, and you can listen to FM radio on Bluetooth, as long as you have a wired headphone plugged in.
I've tried with an audio cable without headphones, and it works; but perhaps because the connector is low-quality, the phone thinks I'm pressing the headphone remote button, which makes the radio randomly pause.
This is specific to Motorola, which seems to record the analogue radio output, and then sends it through Android's audio system as digital audio. The app can also record FM broadcasts. This does sometimes mess up the system audio output, but stopping the radio app Activity usually fixes it.
I also remember that installing CyanogenMod on the Nexus One would enable its FM radio, but there is no recording or Bluetooth functionality.