I think it fits the definition perfectly.

A vanity plate or personalized plate [...] is a special type of vehicle registration plate on an automobile or other vehicle. The owner of the vehicle pays extra money to have their own choice of numbers or letters, usually portraying a recognizable phrase, slogan, or abbreviation, on their plate. https://en.wikipedia.org/wiki/Vanity_plate

One difference is that in Germany you may only pay for the 2nd and 3rd part of the registration number. The 1st part is the area where the car is registered, like you said. Another difference is that in Germany vanity plates are so cheap that they're the norm rather than the exception. The translation for vanity plate is Wunschkennzeichen btw, which is a term that most German speakers are familiar with.