The author didn't write "isomorphisms in metric spaces". Why? Not important? I am reading "Introduction to Set Theory and Topology" (in Japanese) by Kazuo Matsuzaka.
In this book, the author wrote "isomorphisms in topological spaces" (homeomorphisms).
But the author didn't write "isomorphisms in metric spaces".
I want to define "isomorphisms in metric spaces" as follows:

Let $(S,d)$ and $(S',d')$ be metric spaces.
If there is a bijection $f:S\to S'$ such that $d(x,y)=d'(f(x),f(y))$ for any $x,y\in S$, we say that $(S,d)$ and $(S',d')$ are isomorphic.

I wonder why the author didn't write "isomorphisms in metric spaces".
Not important?
 A: The definition you've provided is known as isometry, and such metric spaces are called isometric. This is a very strong condition, and most functions you'll see are not isometries. For example even if you don't assume being bijective, isometries are always injective.
Typically an "isomorphism in metric spaces" is defined simply as a continuous bijection with continuous inverse. These are known as homeomorphisms. Note however that continuity depends on topology only, and probably that's why the author wrote "isomorphisms in topological spaces".
Also note that most homeomorphisms you'll see are not isometries. For example there is no isometry between $(0,1)$ and $\mathbb{R}$ (because one is bounded and the other is not, isometries preserve this property) even though they are homeomorphic. Moreover the famous Mazur-Ulam theorem says that an isometry between normed spaces is an affine transformation, i.e. a composition of linear map and translation. This is a rather strong restriction from the metric/topological perspective.
And in some sense you are right: isometries is the thing that really talks about metric. Homeomorphisms don't care. But in practice it seems that isometries are less useful (the condition is too strong).
