Suppose we have two normed/metric spaces $X$ and $Y$ and suppose $X$ and $Y$ are isometrically isomorphic meaning there exists an isomorphism $T: X \rightarrow Y$ which is also an isometry. This is the "nicest" map possible between these spaces.
However, if one should think of an isomorphism in general as something that "preserves structure", why is isometry property not just contained in the definition of an isomorphism? Wouldn't one expect that distances is part of the structure of a set (with a distance function)?
In Rudin's "Functional Analysis", he also writes "isometric isomorphism" when appropriate. So from I can tell this in the general notion? Can anyone help shed some light on this?
This is, as far as I know, a historical accident. From a category-theoretic perspective it comes out of conflating the category of normed spaces (with maps of norm $\le 1$) with the category of topological vector spaces; it's an isomorphism in the latter category but not the former (and an isomorphism in the former is an isometric isomorphism).
Similarly continuous functions were first defined between metric spaces before being defined between topological spaces, and when we talk about homeomorphisms of metric spaces we are really working with isomorphisms in the category of topological spaces. But metric spaces came first and similarly normed spaces came first.
If you'd like to learn much more about the history here I like Dieudonne's History of Functional Analysis.