It has to satisfy the properties of the identity arrow - i.e. its left and right composition with any other arrow is that arrow.
Of course, you can have multiple different categories where the objects are types (for some definition of "types" - it's actually really complicated, if not impossible, once you start looking at polymorphic or dependent types) and arrows are something somehow related to functions between them. What exactly you choose to call "arrow" is up to you, as long as it satisfies the required properties (composition and identity).
A trivial example, you can say that "let us define that the (sole) arrow between type $A$ and type $B$ represents the family of all functions between $A$ and $B$" - in this category the sole arrow $A \to A$ will happen to also be the identity arrow. I suppose this is what you mean under "looking at type level".
Or you can say: For a given family of equivalence relations $Eq_T$ on every type $T$, let us define that there is one arrow between type $A$ and type $B$ for every function $A \to B$, modulo the function equivalence relation "maps $Eq_A$ to $Eq_B$". Then if you take the usual equality relation on integers and consider the arrows $int \to int$, in this category you'll only have one identity arrow - the function that maps a number to itself. The function $x \to x+1$ will not work.
