It's actually Z80-ish with some features missing and some new ones included. It's a Sharp LR35902 SoC with an SM83 core (not sure if it has been used in non-GB SoCs).

I can recommend The Ultimate Gameboy Talk for details on it (and much more).

https://m.youtube.com/watch?v=HyzD8pNlpwI

EDIT: I had the SoC and the core names the wrong way around. Thanks svendahlstrand.