Is a continuous function like a homomorphism/isomorphism for metric spaces? If I had to define a notion of a homomorphism/isomorphism on metric spaces, I'd say something like this.
Let $A$ and $B$ be metric spaces with norms $\| \cdot \|_A$ and $\| \cdot \|_B$ respectively. A function $\varphi:A \to B$ is called an homomorphism between $A$ and $B$ if $\|\varphi(a)\|_B = \|a\|_A$ for all $a \in A$. Moreover, $\varphi$ is said to be an isomorphism between $A$ and $B$ if it is bijective.
Now, I've never seen such a definition before. Only stuff like continuous functions is discussed in the context of metric spaces. Is this definition somehow obsoleted by (or exactly the same as) the notion of a continuous function?
 A: This definition is too restrictive and you are also mildly confused. First, not every metric space is a normed space. Second, in practice we want to consider many maps between metric spaces that aren't isometries. The "correct" definition of homomorphism between metric spaces, for reasons that I won't get into here (but see this blog post for my thoughts on this issue), is a short map or weak contraction, which is a map $f : M \to N$ satisfying
$$d_N(f(a), f(b)) \le d_M(a, b)$$
for all $a, b \in M$. This definition has the desirable property that two metric spaces are isomorphic iff they are isometrically isomorphic in the usual sense, and it is also much less restrictive than asking for an isometry.  
A somewhat less restrictive notion is that of a Lipschitz continuous function. This is useful as well. Two metric spaces which are isomorphic in the sense that there are two Lipschitz maps between them that are inverses are said to be bi-Lipschitz equivalent, which is a weaker notion than isometrically isomorphic but is useful in some settings; it is related to the notion of quasi-isometry which appears in geometric group theory. 
An even less restrictive notion is that of a uniformly continuous function. This is the correct notion of homomorphism between uniform spaces. 
The least restrictive notion, that of a continuous function, is the correct notion of homomorphism between topological spaces. Strictly speaking, when you talk about continuous functions between metric spaces, the category you're constructing is really the category of metrizable topological spaces. 
A: The word you are looking for is "isometry". 
A: You are confusing metric spaces with normed spaces. A metric space is a set $X$ together with a function $d:X\times X\to R$ satisfying 


*

*$d(x,y)=0 \iff x=y$

*$d(x,y)=d(y,x)$

*$d(x,z)\le d(x,y)+d(y,z)$
Then, the concept of a structure preserving function (i.e., homomorphism) is (somewhat arguably) a function $f:X\to Y$, where $X$ and $Y$ are metric spaces with their $d$(istance) function, such that $$d(f(x),f(x'))\le d(x,x')$$ for all $x,x'in X$. Such mappings are caled nonexpansive or short. The concept of an isomorphism is then that of an isometry: a homomorphism $f$ which is bijective and satisfies $$d(f(x),f(x'))= d(x,x').$$
A: Continuous functions are the morphisms in the category of topological spaces (and as well in the subcategory of metric spaces) just like homomorphisms are the morphisms in the category of groups (for example).
Note that an isomorphism (called homeomorphism for topological spaces) is not simply a bijective continuous map, but rather a map with a continuous inverse (i.e. we have morhisms $f$, $f^{-1}$ such that $f\circ f^{-1}=\operatorname{id}$ and $f^{-1}\circ f=\operatorname{id}$). Again, this notion is "the same" for metric spaces. However, a map that preserves the metric, $d(f(x),f(y))=d(x,y)$, is more than just a homeomorphism; this stronger notion that really makes use of the metric itself and not just of the notions of convergence etc. It induces an isometry and has no counterpart in the category of groups.
