Gameboy is really slow when compared to modern hardware. One ECDSA signature that takes around 100ms on Trezor (32-bit CPU @ 120 MHz) would take around a minute on GameBoy (8-bit CPU @ ~4 MHz.
If we're talking about GameBoy Advance (32-bit CPU @ 25 MHz) - this more plausible but still 5 times slower than Trezor.
I think waiting a full minute is a perfectly justifiable trade-off for a dystopian future that at least forces people to have an iconic '80's handheld strapped to their hip. If we're headed toward a dystopia, we can at least try to set the stage for a certain standard of aesthetics.
It could perform it, but as the other comment has outlined it would be very slow (and not very secure)!