The cartridge would indeed be Hardware, possibly with some bit of firmware or software storage are for game saves. Technically it might be entirely Firmware if

The cartridge would indeed be Hardware, possibly with some bit of firmware or software storage are for game saves. Technically it might be entirely Firmware if the manufacturers decided to just use a single EEPROM chip for the whole thing, even if during normal operation the software on the chip only ever re-programmed a single page.