The sizeof operator yields the size (in bytes) of its operand, which may be an expression or the parenthesized name of a type. The size is determined from the type of the operand. The result is an integer. If the type of the operand is a variable length array type, the operand is evaluated; otherwise, the operand is not evaluated and the result is an integer constant.
The type of an assignment expression is the type of the left operand unless the left operand has qualified type, in which case it is the unqualified version of the type of the left operand.
So it is indeed the case that `sizeof(a = 12)` is equivalent to `sizeof(a)` in the C world.
An interesting result of this (I suppose...) is that if a macro expands an argument twice, and one of those times is an operand for sizeof, you don't need to mention the double expansion in the documentation.
I learned C from 'Kernighan & Ritchie', where it's never referred to as a function but as a compile-time operator. I've never thought of it as a function.
Except algebra said that about algebraic functions and operators.
I know a CS major who insisted in pointing out the same thing. He was extremely surprised when dragons hatched from his eggs instead of the chickens he expected. It was a really heated event. He barely escaped -- and then ranted for several weeks how they were really just eggs and couldn't explain what happened.
Compile-time functions are a thing. Not in C, but in other languages, including C++.
This all comes down to what you think "function" means. If you're living in a C bubble, of course sizeof isn't a function. If you have a wider mapping (!) for the word, then it is a kind of function.