Does anyone know of a product such as 2 way adhesive that can be used to attach a rear license plate without drilling holes?

Industrial Velcro- sold at Home Depot and Ace HW stores....

you can buy rubberized magnetic bars, with mounting holes - made just for this purpose on amazon

no experience with them, but the idea is solid

assuming your bumper has the typical hole dimension i would say to mount a us plate i got a universal one off ebay that has proper spacing to mount holder to bumper and then 4 small screws to hold license plate to holder. if you dont have any holes at all then i guess the 2 suggestions given above will work too

3M automotive trim adhesive tape (double-sided) comes in various widths and sticks incredibly well. It is intended to be "permanent", though.

What does your bumper look like,? They all had holes for mounting a plate, front and back, from the factory.

Tow hook license plate mount. No drilling required, just screw it in. Click to view attachment

The bumper is being fabricated locally and has a smooth surface. I don't have a photo but was hoping to have it retain an undrilled look in case I take pictures of it without a license plate or frame in place. I appreciate the ideas and suggestions.

Add the holes and have chrome plugs for picture time. Or fabricate a fixed welded on nut behind the bumper to attach the plate holder. Like a trailer hitch.
I'm not an artist either.