I know of course where the vehicle registration office (or what it is called) for cars is located, along the Hang Dong rd.
But what about motorcycles? I will probably buy one in BKK and must, I assume, then change the registration plate to a CM one. And maybe register a repaint in another colour. It used to be somewhere by the river, but I haven't had a bike for ten years, so I wonder where it is situated now. Even if it is on the same place I can't really remember exactly where. An address and a road description would be very much appreciated.

Google is your friend -

https://www.google.com/maps/d/u/0/viewer?mid=z34UOjQnWNW0.k8kRbO0KWyWo&hl=en_US

There is also one on the north bank of the Ping river. Google Holiday Inn and it is right across the bridge.

There is no legal requirement to change from a BKK plate to a Chiang Mai plate. Mine has been the same, and been renewed, in Chonburi, Hua Hin, Krabi, Rayong, Chiang Mai

You'll need to go to the Provincial Land Transport Office in Nong Hoi- it's on Google Maps.

A tip.If you haven't ridden A motorcycle in 10 years be careful.Chiang Mai is getting more crowded every day.Not hard to get ran over.