While different layers may use words of the same size, there are still differences, for example what is valid and what is not. While for example U+00D800 is a perfectly fine code point, the first high-surrogate, 0x0000D800 is not a valid UTF-32 code unit. 0xC0 0xA0 is a perfectly fine pair of bytes, both are valid UTF-8 code units, and they could become the code point U+000020 if only 0xC0 0xA0 were not an invalid code unit subsequence.
So yes, while I agree that UTF-16 is special in that sense that one has to deal with 8, 16 and 32 bit words, I don't think that one should dismiss the concept of code units for all encoding forms but UTF-16. There enough subtle details between the different layers so that the distinction is warranted. And that is actually something I really like about the Unicode standard, it is really precise and doesn't mix up things that are superficially the same.