Simple answer is that the FB messenger app does not have support for AA built-in. I agree this seems like an obvious feature to include and is not technically difficult.
Considering the number of people who use Facebook (i.e. friends and family who are not technically savvy) it would be a good addition. I thought any app conforming to standards of SMS would naturally be picked up. Perhaps FB Messenger is not using SMS???
Based on what I've read the app has to be adapted for messaging with Android Auto, independent of whether it's using SMS (which I think Facebook messenger can do). WhatsApp doesn't do SMS but it does work with AA