Whether or not a person can look up another person's license plate largely depends on the state where the vehicle is registered. Some states allow looking up full license plate and owner information, while others only allow someone to confirm that the license plate exists.
Some states allow no license plate searches by private citizens whatsoever, such as Oregon. It is not possible to look up another person's plates in Oregon without contacting law enforcement, even to determine if a potential vanity plate is in use. However, it is possible in some states to indirectly look up personal information by using a license plate number to obtain a vehicle identification number, then using that number in subsequent searches.