Apparently, six different types were announced at the start (all having a Super Mario Bros. theme) but more were released from different games, like Zelda and Punch-Out! as well.
Originally, they sold for a whopping $1.99 each, and I assume they were meant as “trophies” to be given to your kid after they showcased some amazing Nintendo skill. On the base is a sticker where you can put your name, the high score and the date you achieved the score; a replacement sticker also came in the box should you beat your own record.
They show up a fairly frequently, usually sealed more often than not. The price range is all over the place for them, but this one that ended recently was a pretty good deal in my opinion ($99.00 for all five of the Mario ones).
Anyway, this seller has two sealed ones up for sale. They have a Punch Out! “Bald Bull and Mac Mix it Up”, and a The Legend of Zelda “A Trap Attacks Link”, both from 1988.