The Z80 fills some redundancies and gaps in the 8080 instruction set and adds the concept of prefixed instructions (via the CB, DD, FD and ED prefix bytes), but is otherwise compatible with the 8080 (it can run 8080 code which doesn't use any of the redundant opcodes which have been repurposed in the Z80).
Looking at the Gameboy CPU instruction matrix (https://www.pastraiser.com/cpu/gameboy/gameboy_opcodes.html) versus the 8080 matrix (https://pastraiser.com/cpu/i8080/i8080_opcodes.html), the Gameboy CPU looks a lot more like a Z80 than an 8080. For instance the Gameboy CPU has the conditional relative jump instructions (like JR NZ, JR NC, ...) of the Z80, while the 8080 only has redundant NOPs in those instruction slots.
Also while the Gameboy CPU is missing the DD, FD and ED prefixed instruction ranges, it has the complete CB-prefixed 'bit twiddling' instruction range of the Z80).
So IMHO it's more correct to call the Gameboy CPU a stripped down Z80 than an extended 8080 (but of course the Z80 is also "just" an extended 8080, so it's also not wrong to call the Gameboy CPU an extended 8080, the 8080 is basically the grandfather, and the Z80 the father).