If you want to print one, maybe you should outsource it (let it print the tag on both sides), even the most affordable printers are in the \$100 - \$150 price range. If you want a printer and use it also to create ID tags, you could go for an FDM printer. Considering your request of having the tag inside (and through) the ID-card you need a dual filament option (one or two nozzle arrangement). If the tag can sit on top you can print it with a filament change with a single filament single nozzle printer. But, don't expect to get crystal clear prints (see experience printing signs below)!
As an alternative, you could print a blank PLA ID-card and laser mark the tag onto both sides, see e.g. this video.
If it is a small batch you can also consider printing/lasering stickers and stick these onto blank ID cards.
I've done some signs with black letters on a white background for "on-lay", inlay and through arrangements using a more expensive (for home use) dual extruder 3D printer (Ultimaker 3 Extended about \$5000,-) with PETG, but the results were not very satisfying. Usually the black smears out on or in the white no matter tweaking the options. Considering the size of an ID-card, the amount of tag squares, this is even more likely to happen when you print at that small size (the signs I printed were sized similar to the "A5" paper standard).
From my experience I would say that a 3D printer may not be the best solution for your task.