Word for a person who's in possession of something, but doesn't necessarily own it

Solution 1:

The word "custodian" may be what you're looking for, although a little formal. Having custody of a vehicle would imply having the keys and a responsibility to take care of it on behalf of the owner, but not necessarily the legal ownership.

Alternatively, if you don't need to describe it in terms of the vehicle, you might want to refer to this person as the "representative" of the owner/seller (who may be the owner/seller themselves).

Solution 2:

You may use "holder"

"He's the holder of his brother's car."

examples:

"The second issue is who is the holder of the privilege."

"If any person is the holder of a negotiable instrument the value of which has been paid..."

"it is the holder of the Crown who possesses such powers."

"She is the holder of a EU passport" - a person who has possession or control of something.