Why is sizeof considered an operator?

It can be used as a compile-time constant, which is only possible if it's an operator rather than a function. For instance:

union foo {
    int i;
    char c[sizeof(int)];
};

Syntactically if it weren't an operator then it would have to be a preprocessor macro since functions can't take types as arguments. That would be a difficult macro to implement since sizeof can take both types and variables as an argument.

Because the C standard says so, and it gets the only vote.

As consequences:

• The operand of sizeof can be a parenthesised type, sizeof (int), instead of an object expression.
• The parentheses are unnecessary: int a; printf("%d\n", sizeof a); is perfectly fine. They're often seen, firstly because they're needed as part of a type cast expression, and secondly because sizeof has very high precedence, so sizeof a + b isn't the same as sizeof (a+b). But they aren't part of the invocation of sizeof, they're part of the operand.
• You can't take the address of sizeof.
• The expression which is the operand of sizeof is not evaluated at runtime (sizeof a++ does not modify a).
• The expression which is the operand of sizeof can have any type except void, or function types. Indeed, that's kind of the point of sizeof.

A function would differ on all those points. There are probably other differences between a function and a unary operator, but I think that's enough to show why sizeof could not be a function even if there was a reason to want it to be.

Because the C standard says so, and it gets the only vote.

And the standard is probably correct because sizeof takes a type and

In general, if either the domain or codomain (or both) of a function contains elements significantly more complex than real numbers, that function is referred to as an operator. Conversely, if neither the domain nor the codomain of a function contain elements more complicated than real numbers, that function is likely to be referred to simply as a function. Trigonometric functions such as cosine are examples of the latter case.

Additionally, when functions are used so often that they have evolved faster or easier notations than the generic F(x,y,z,...) form, the resulting special forms are also called operators. Examples include infix operators such as addition "+" and division "/", and postfix operators such as factorial "!". This usage is unrelated to the complexity of the entities involved.

(Wikipedia)

Because it's not a function. You can use it like that:

int a;
printf("%d\n", sizeof a);

Function does have entry point, code, etc. Function is to be run at runtime (or inlined), sizeof has to be determined at compile-time.