So I've been having this exact problem with my Jaguar XE (2015) and my iphone 8 ever since upgrading to iOS 13. I have tried all manner of fixes. Deleting Bluetooth connections, repairing the car to the phone and nothing worked. I found that audio will play fine in the car if connected by USB to the car and after twenty minutes of this the connection would be stable again for bluetooth, but who wants to be doing that all the time to listen to music or make phone calls whilst driving...
I booked my car in with my local dealership and ended up paying an annoying amount of money to have the cars firmware updated to be on the latest version as that was the only thing I could think of left to do was the cars bluetooth software. Since Jaguar updated the cars firmware with the latest version my iphone is now connecting find for music and calls and haven't had a drop since yesterday. Annoyingly Apple released 13.1.2 (Addresses an issue with bluetooth connecting to cars) after I paid money to get my car updated....
In iOS 13 apple changed it's bluetooth privacy settings, which is why some people may have noticed some apps asking for bluetooth access. I think it's the same thing with Cars being paired to the phone, it's requesting something the car can't agree to.
Please let me know if iOS13.1.2 fixed these issues for you all.