When you get an MMSI number from the cognizant administration, the FCC in the US, it is registered in the ITU (International Telecommunications Union) database. Anyone can search the basic database that will give you the vessel name, callsign, country of registry, etc. I'm sure the restricted version available to administrations has more info. So, yes, the Coast Guard can get your Canadian info. If you get the MMSI number in the US from BoatUS or someone like that, the info is not registered with the ITU. That is why it is for domestic use only. I just checked as our sailboat has an FCC issued MMSI and our C-Dory does not have an FCC license and I got the MMSI for that vessel from BoatUS. Our sailboat shows up in the ITU database. Our C-Dory does not. You can search the database here:
https://www.itu.int/mmsapp/ShipStation/list
https://www.itu.int/mmsapp/ShipStation/list