I don't think so. Not entirely. Yes, the Apple II has a simple (though rather eccentric) video system that is easy to emulate, but I'm 99.9% sure that this emulator is using an interpreter to execute Apple 6502 code. I hypothesize this model for the emulator's operation:
The 6502 code in interpreted. No big trick; interpreters for 6502 code were built into every 6502 monitor from back in the day; this is how the monitor was able to "STEP" through sections of code and display the contents of the registers after each instruction execution. The emulator's interpreter would execute code at maybe 1/10 the speed of actual direct execution by the CPU, and would interpret native Apple II machine language software including the Apple's ROMs (which include the Integer BASIC interpeter).
Since the Atari and Apple both need zero page memory, and each needs its hardware memory maps in a specific fixed place; so pretty much all of Apple's memory would have to be virtualized into new locations in the Atari's memory; this is why the Apple's machine code need to be interpreted instead of executed directly.
A portion of the Atari's memory would contain the emulated Apple II's memory, but another portion of the Atari's memory would have to contain the Atari's screen maps. The machine code interpreter would have a list of memory locations that have input/output functions in the Apple II; whenever virtualized Apple memory is written to, this list is consulted; if, say, part of the screen's memory map is changed, the emulator pauses in the interpretation of code, and updates whatever part of the Atari's screen needs to be changed. Likewise, the emulator would know how the Apple's keyboard, paddle controllers, etc. are mapped in memory so that when this memory is referenced, the correct information can be read from the real Atari and transliterated to the virtual Apple.
And the whole thing is slow. Ten or more times slower than a real Apple II.
So the Apple II and Atari having the same CPU is not a big factor in the feasibility of this emulator; an interpreter can be written to run any instruction code on any CPU. Maybe someone will write an Apple II emulator for the TRS-80 Color Computer.
Now I just read somewhere else that in the early 1980's, a software company developed VIC-20 games on an emulator that ran on an Apple II. I bet that the VIC's graphics screens really looked like crap on the Apple's HirRes screen.