Product placement in video games has been around almost as long as video games. However, for at least a short time, it was even popular to build games around a product. Cool Spot is a game released initially for the Super Nintendo and Sega Genesis in 1993 based around a mascot for 7-Up.
Given the typical quality of licensed products you would expect this to be a truly awful game. However, it really wasn't. Your goal is to rescue other Spots throughout each level with a typically increasing number to rescue as you get further in the game. Your weapon is soda bubbles and you also have the ability to climb and stick to things.
Cool Spot got reasonably good reviews and was later ported to the Sega Master System, Game Gear, Game Boy, Amiga and DOS in 1994. This one is worth a try for the novelty factor alone but you'll have to track down and original copy or use emulation as this one hasn't been re-released as far as I know.