Why do we study the number of homomorphisms/isomorphisms between fields?

In a basic sense, algebra is about sets that have certain kinds of structures and about the functions that preserve those structures—namely, morphisms.

Morphisms reveal the structure of spaces. For example, if you know about the properties of one space $X$, the morphisms into another space $Y$ may tell you about $Y$.

As for why isomorphisms might not be as relevant: one way to look at it is that isomorphisms preserve too much structure to reveal interesting information about a space's structure —because if there is an isomorphism between two spaces, they are actually identical as far as their algebraic structure is concerned.

I am not sure why the number would be an especially important property to know about spaces, except that no matter what space you're looking at, you can always ask how many morphisms there are. The number of isomorphisms is interesting because it shows something about the symmetry of a space— how many ways one space can be mapped onto another.