There's a real "Surf Ninjas" game for the Sega Game Gear, but it's completely different from what Adam was playing. However, the game Adam plays in the movie is also real. It was developed by Sega exclusively for use in the movie only.
This film was made as a vehicle for Ernie Reyes, Jr.. Because New Line Cinema (same studio who made the original 3 "Ninja Turtles" movies and "Surf Ninjas") saw potential in Ernie Reyes, Jr. after As Tartarugas Ninja II: O Segredo do Ooze (1991) opened to good numbers, he was given a three-picture deal, with this being the first. Following this film's failure at the box office, his deal was terminated.
The portable video game system Adam plays throughout the movie is the Sega Game Gear, a handheld version of the Sega Master System. It was Sega's answer to the Nintendo Game Boy, which came out a year earlier.
Leslie Nielsen brought a fart machine to set and would mess around and prank the crew with it. He did this on nearly every movie set he was on.