I can't find a solution so I'm assuming this hasn't been acknowledged or fixed :-(
I've used CarPlay without any issue at all for about 4 years, most recently with an iPhone XS. I use it almost exclusively for audiobooks and maps, though occasionally also Applel Music.
About two weeks ago I got a new phone - iPhone 13, brand new - and CarPlay wouldn't work. I couldn't get it going at all initially. Then two days ago I somehow managed to get it to connect. (At some point in the process I saw a message flash across my iPhone screen suggesting that it was updating the CarPlay software. Since there doesn't seem to be any for the phone I'm now guessing that meant in the car...? Skoda Fabia.) Anyway, next journey (following morning) it just didn't work again. CarPlay logo came onto car screen and almost immediately it said it was disconnecting the iPhone.
I tried wifi but it's very laggy and so annoying that it's not useable for me.
I keep looking for a solution on the web but there doesn't seem to be one. Having read this and other threads I don't think it's about the model of iPhone, or about the cable or the car.