The technology is available. It's called IPFS. (interplanetary file system). And you don't store all data on the blockchain, you just put the transactions there.
Storing all the data would be meaningless i guess.
Yeah I think it won't be necessary, it won't be beneficial to anyone, it will just be a waste of space. I think what's really important to store are the transactions and the information about the vehicles and their owners. Speaking of owners though, what information from the owners will be stored in the blockchain? And will the owners need to present any documentations to prove that they're really the owners of the cars? And how can users register on the blockchain do you require any proof of identification?
As for the registration on the blockchain, I don't really understand what you mean. Blockchain contains all the info about what happened with a car during it's lifecycle, the users (if you mean just the people who wants to get info about the car) don't need to register anywhere, they could get this info for free whenever they want.
But there are name collisions. There are so many people with the same name. If you want to know the history of ownership of a second hand car for whatever reason, for example an accident or theft, would a unique public ID not be better than a name?