Does every binary operation have an identity element? If not, then what kinds of operations do and do not have these identities?
For example, $1$ is a multiplicative identity for integers, real numbers, and complex numbers.
Based on the clarifying edit, I think what you're really trying to ask here is whether all binary operations have an identity element. ("Function" is a broad term that includes functions that only take one input. "Functional identity" is not, as far as I know, a standard term in the sense in which you're using it in the comment.)
The answer to your question is no. For example, let the binary operation \$ on the real numbers as be defined as $x\$y=|x|+|y|+1$. Then there is no left or right identity element for \$ (because we always have $|x\$y|>|x|$ and $|x\$y|>|y|$).
You can also have binary operations that have a left identity but not a right identity, or vice versa. For example, let $x\%y=|x|+y-7$. Then $7\%y=y$ for any $y$, but there is no number that plays the same role as 7 on the right-hand side.