For me this is analogous to, say, a PC running a NES emulator.

You can talk about the emulated CPU. Its state: its 8-bit register values, the contents of its 2KB of RAM, what 6502-architecture code it's currently running. Its concept of time: how many clock cycles it spends doing what, at an assumed fixed number of nanoseconds per clock cycle. Its inputs and outputs: controller button presses, video out.

Or you can talk about the PC's CPU. Its 64-bit register values, the contents of its gigabytes of RAM, what x86-architecture code it's currently running. Its own clock cycles, its own inputs and outputs.

Both of those can be considered to exist at the same time.

But of course, the emulated CPU doesn’t have an independent physical existence; it’s just an abstraction that exists within the PC. Its register values and RAM contents are represented by some part of the PC’s register values and RAM contents, with some arbitrary encoding depending on the emulator in use. Its time flows whenever the PC feels like running the emulation. The emulator might be set to 1x real time, or 2x or 0.5x, but even that setting only applies on average; individual clock cycles will always proceed at a highly erratic rate. The emulated CPU’s output might go to a real screen or it might just be saved as a video file. And so on.

But if the CPU isn’t real:

(1) Does that mean it’s a “p-zombie” that is only pretending to run 6502 code, but isn’t really?

(2) Does that mean you’re not really playing Super Mario Bros. if you play on an emulator?

My answer to 1 is: maybe, maybe not, but it makes no difference. Because my answer to 2 is: no, you definitely are playing the same game regardless of whether you’re using an emulator or not. The essence of what it means to “play Super Mario Bros.” is to interact with a system that follows certain rules to map controller inputs to video outputs. The rules are a mathematical abstraction, not inherently connected to any physical object. So it doesn’t matter whether the rules are implemented by a physical CPU or an emulator inside another CPU.

And I see consciousness as basically the same thing. A conscious being, to me, is fundamentally a mathematical abstraction consisting of some state, some inputs and outputs, and rules for manipulating the state and producing outputs from inputs, where the rules have to meet certain standards to count as conscious. For example, the rules should be able to perform general-purpose logical reasoning; they should include some kind of concept of self; etc. The exact boundaries could be litigated, but at minimum the rules corresponding to the operation of the human brain quality.

And a physical brain is really just one possible physical representation of that abstraction. A Chinese room would work too. It would be impossibly slow, and the person would need an astronomical number of filing cabinets to store the necessary data, and they’d inevitably commit calculation errors, but aside from that it would work.

So yes, the Chinese room, or the mathematical process it represents, can have consciousness, while the person within it has their own independent consciousness. Just as a NES CPU can exist “inside” a PC CPU, an emulated brain can exist “inside” a real brain (well, “inside” except for the filing cabinets).