can any FM synthesizer with N number of operators enabled likewise make the exact same sounds as any other FM synthesizer with N number of operators enabled (e.g. Ableton's Operator vs. FM8), can any wavetable synthesizer with N number of oscillators make the exact same sounds as any other wavetable synthesizer with N number of oscillators, and so on...? Why or why not?
My reading of this is that "and so on" means you have the same question about other types of synthesizers beyond FM and wavetable. If that's true, then I can speak to analog synthesizers. It is absolutely not a given that all analog synthesizers can make the exact same sounds. It's not even true that all analog subtractive synthesizers with three oscillators and a four-pole low pass filter can make the same sounds.
This is because in analog synthesizers, circuit design matters. Even from the same company, different oscillator and filter circuits will sound different. I have four analog synthesizers made by Moog. Two of them seem to have the same or similar oscillator circuit designs and they sound quite similar. The other two have different designs and sound close to each other but quite different from the first two. Once we consider other products, such as Sequential Circuits, Korg, or Arturia products, the designs differ by even greater degrees and the sonic differences are much more obvious.
I don't quite understand this part:
Please do not consider non-sonic differences such as interface differences
Any design difference can have a sonic effect, so I can't figure out what a "non-sonic" difference would be.
I also want to mention that the quoted definition of Turing Completeness uses the word "approximately", but your question uses the word "exactly". Subtractive analog synths with the same number of oscillators and four-pole low pass filters can make approximately the same sounds as each other. There are other analog synth designs, such as subharmonic synthesizers, and the variation in components and internal patching usually means each synth has its own capabilities, which is why people will buy a variety of otherwise similar analog synths.