I'm guessing you are playing in your web browser. This is not actually an issue with the game, but with your browser clearing cookies. The web Gameboy emulator on itch uses cookies to store your save data locally on your machine, so every time your cookies get cleared your save goes with them. Similarly, if you play in incognito mode your browser will clear all cookies at end of session anyway. Check the privacy settings on your browser!
Alternatively you can download one of the ROM files listed above and play TaiFab on your own emulator or flash cart setup and have more direct access to the save files.
If you are not playing in browser, please tell us more about your issue!