Plaque tape is designed to stick plaques/trophies together. It is very strong. If you use it to hang your nameplate or holder, you will have a hard time getting it back off the wall/door without removing paint or more.
Foam tape is also strong, and easier to remove from a wall/door without damage. Downside? it is 1/16th" thick and pushes your sign/holder off the wall that much.
Velcro-type Hook and Loop and Rubberized Magnetic Tape are also available.
Our custom signs usually come with plaque tape on the back.
Our Wall/Door Name Plate Holders have holes for screws (not provided or recommended),
but if you want to use screws to mount them, 1 inch drywall screws work well (1/4" head tapered to 1/8" hole).