That's an old trick (though no less cool). The GBA had a GBC cpu in it for backwards compatibility, and it used a physical switch to change modes, toggled by the cartridge itself. And iirc the PS2 reused the PS1 cpu as a component on the board for I/O, so it was available for backwards compatibility as well (though with emulation help for other components).
But that button is also responsible for making the cartridge bus actually function with GB/GBC cartridges. If you switch into GBC mode without the button being pressed, nothing can be read from the cartridge. (Normally the register bit that switches to GBC mode is locked out and only the BIOS can write it, but if you are executing code in the BIOS, a timed DMA transfer can perform that write.)
Which has some interesting implications for softmodded systems. I play my PS2 games via Ethernet off a Raspberry Pi Samba server—it’s incredible to me that it’s possible and works so well, but because it’s using the PS1 CPU for I/O you can’t use it for PS1 games.