I think this is intended to be a feature eventually, possibly even by the next stable release, though I can’t be sure.
The issue right now is that dogs don’t have the same abilities and options players and NPCs(?) have (I haven’t actually tried getting an NPC into a car before, I play without them but I assume they can do it). Without the ability to “sit” at a seat they’re treated as an object separate from the car when it starts accelerating, and usually get 'sploded shortly afterward.
Work is being done on friendly dog inventories and stuff I believe, so dogs riding in cars is probably not far away.
