Emil1000
These devices use what are called pairing records to store information about the connections and devices. On most headsets, there are no records - the pairing information is stored on the host (iPhone in this case). Car kits are different - most car kits also store pairing records. When you select 'Forget This Device" on the iPhone, it deletes the pairing record from the phone. What it doesn't do is remove any of the records on the car kit device. The result is that the car kit belives it has already 'advertised' to the phone, and doesn't send out advertising connection information. As such, the iPhone doesn't know there is a 'new' device.
This happened to me on my Audi A6. Once I reset the hands-free settings on the car, the pairing auto-magically showed up again allowing me to reconnect. I'd advise you to see if you can locate the car device settings that allow you to reset the bluetooth connections, then try it again.
Best,
CJ