It dates back to the very first automobiles, which often were partially constructed out of wood.
Towards the later part of the century, many automakers began using various types of fake or synthetic wood to cut down on costs.
[1] Beginning around the late 20th century, wood trim began to fall in popularity, being replaced by plastic, carbon fiber or aluminum composites in many cars.
Wood trim is still offered on many luxury cars, and third party kits consisting of veneer can be found for retrofitting wood trim into cars that did not come with it.
Wood trim has seen a slight resurgence in recent years with the use of sustainably sourced and reclaimed wood by automakers looking to appeal to environmentally conscious buyers.