Hi Jasmine,
The issue is not with the console, but with the game itself. Certain gameboy games, especially gameboy color games, contain save files that are reliant on a soldered-in battery within the cartridge. This means that if the battery inside the cartridge dies, the board will lose its ability to retain and overwrite a save file. Don't worry, though! This is a very common issue with these games as they are 10-30 years old, and their battery life is estimated at 5-9 years in most cases. Dead save batteries are especially prominent in games like Pokemon Gold and Pokemon Silver, due to the internal clock function these games possess. The internal clock tends to drain the battery over time. But other games are just as prone to losing their ability to save over time. There is an easy remedy to this, though. Simply replace the battery in the cartridge, and you will be able to enjoy your games again! To do this, however, you must have a soldering iron and a 3.8mm security bit (tool used to open the cartridge). The battery inside of the cartridge can be replaced in as little as three steps:
1. open the cartridge
2. desolder the battery contacts from the board (note: do not make contact between the battery and the soldering iron for obvious safety reasons)
3. install the new battery by soldering its contacts onto the board
After following these steps, just reassemble the cartridge and enjoy your game! In case you cannot find the proper CR2025 batteries and the appropriate bit, I have included a link below for the items. I, personally, have shopped from this seller and can vouch that their stuff is legit. They do take a dreadfully long time to ship, so I would shop around for a similar listing if possible. https://www.ebay.com/itm/X5-Game-Boy-GB-Save-Game-Battery-CR2025-Tab-3-8mm-Security-Screw-Bit-Pokemon/201824589815?ssPageName=STRK%3AMEBIDX%3AIT&_trksid=p2057872.m2749.l2649 Happy fixing!