You got me curious as well so I googled it and it looks like CarPlay just uses the screen as a monitor with no messages or anything downloaded:
https://discussions.apple.com/thread/252600482
Now I wonder what kind of system these vehicles have that downloads text messages. Is that a function of the Bluetooth connectivity or is it a vendor application?
I’ve been bitten by this. I’ve gotten a battery warning in the AM at the start of a day and had my mouse die in the early afternoon. It’s annoying that I can’t just charge it while working so I had to dig around for another mouse and use that while it charged. I hate that I have to check the charge on it and then forget after a while because it does last a long time. So it’ll die on me and I’ll get good about checking charge for a few days and then forget for a while and boom it happens again. If Apple sold this config I would definitely replace my current mouse with a better design.