iPhone in iPod Mode in Car Switches to Different audio App Intermittently
My knee-jerk reaction is to say: This sounds to me like a possible problem with the cable or the Lightning port on the phone.  It's as if the phone is getting some kind of remote control command via the cable.  I've seen similar skipping behavior with a bad Apple earbud cord.

That being said, I have an iPhone 6s and a new Nissan NVP 3500 that I too connect to the car via USB.  If I'm not playing anything at the moment, it will start playing the first thing alphabetically in my library.  If I'm playing something already, or if the two were connected via Bluetooth before I connect the cable, it will continue as if nothing was wrong.

On a quirky note, sometimes the stereo and the phone will haggle for a bit, then the phone will play the audio through its own speaker and the car will give up trying to access the phone in iPod Out mode.  In that case, I have to reboot the phone altogether to get it to play nicely with the car.  I chalk that up to running the iOS 10 betas.