My Continental head unit does something like this. I read this useful info on the Bergvill F/X download about this problem:
Wrongly connected power supplies
If you connect the Continental radio to the original Porsche radio harness, the radio will work when the ignition is on. However, it will behave as the ignition is turned off: You need to manually turn on the radio every time you want to use it, and it will automatically turn off after 20 minutes. The head unit will also default to FM reception mode when turned on, and will lose any memory of DAB frequency plans. (FM station pre-sets are stored)
You will also experience problems when trying to turn on and off the radio. It may not respond to any attempts to turn it on some time after it has turned off, both manually and automatically.
Correctly connected power supplies
If the switched and permanent power supplies are connected to the radio, it will automatically shut on and off with the ignition. Before shutting off, the radio will show a row of <<<<< characters in the display. The head unit will continue in the same mode of play as it was when turning off the ignition.
I guess you can replace Continental with Kenwood as it sounds like a similar issue.
https://www.bergvillfx.com/categories/downloads