Hello Everyone;
After I upgraded my XS Max from 12.4.x to 13.2.3 I too experienced the same problem with the messages being marked as read after my car notified me of "incoming message from ..."
The forget device/phone reset/re-pair to car did not work for me, so I did the following:
I have a 2016 Subaru Legacy (no car play support).
(while car stereo is on and paired with phone)
- Went to car stereo settings and choose Phone
- Went to message settings
- un-checked "Update message read setting on phone"
Once I did that, everything works as it did in IOS 12.4.x The incoming message is announced via the car prompt, the message notification banner slides down, then goes away. Message app shows badge of unread message. I used Siri via the car and had the message read to me, (I chose not to reply), so Siri ended the sequence of events. Read message badge disappeared.
Repeated several times and all were successful in keeping the "new message" badge displayed when appropriate.
I hope this helps with others! Good Luck!