Is it ok to say that a map $f: X \rightarrow X$ is identical, to mean that it is the identity map?
I believe this terminology would allow one to write statements such as
"The restriction of $g: X \rightarrow X$ to $X' \subset X$ is equal to the identity map on $X'$"
with much less text., i.e.:
"The restriction of $g: X \rightarrow X$ to $X' \subset X$ is identical"
and remaining "smooth" i.e. without using too many symbols (this is a subjective notion but I hope it is understandable).
For me this intuitively doesn't sound quite correct, but a google search shows a few authors have used this phrase and my intuition on such matters has been wrong in the past.
Therefore, I have decided ask here to get a more general consensus. I would also appreciate someone could point out a textbook where it is used (although I doubt this somehow).
I don't think so. This would be incorrect terminology... You say a map is the identity when $m(x)=x\;\forall x \in X$.