I found some search results about Texas Instruments' digital signal processors using 16-bit bytes, and came across this blogpost from 2017 talking about implementing 16-bit bytes in LLVM: https://embecosm.com/2017/04/18/non-8-bit-char-support-in-cl.... Not sure if they actually implemented it, but that was surprising to me that non octet bytes still exist, albeit in a very limited manner.
Do you know of any other uses for bytes that are not 8 bits?
For "bytes" as the term-of-art itself? Probably not. For "codes" or "words"? 5 bits are the standard in Baudot transmission (in teletype though). 6- and 7-bit words were the standards of the day for very old computers (ASCII is in itself a 7-bit code), especially on DEC-produced ones (https://rabbit.eng.miami.edu/info/decchars.html).
NXP makes a number of audio DSPs with a native 24 bit width.
Microchip still ships chips in the PIC family with instructions of various widths including 12 and 14 bit however I believe the data memory on those chips is either 8 or 16 bit. I have no idea how to classify a machine where the instruction and data memory widths don't match.
Unlike POSIX, C merely requires that char be at least 8 bits wide. Although I assume lots of real world code would break if challenged on that particular detail.