Apart from fields, what other mathematical objects have "naturally" as morphisms maps which are all injective?
One way to define fields is as precisely the commutative rings which have no nontrivial ideals; in other words, as precisely the commutative rings which have no nontrivial quotients. You can play this game in other categories too. For example, the analogous subcategory for groups is simple groups (the trivial group is not simple), and the analogous subcategory for rings is simple rings.
Replacing "injective" with "monic" it is of course straightforward to start with a category and restrict to the subcategory of monomorphisms, but presumably this isn't in the spirit of the question.
An example which may or may not be in the spirit of the question is the category of metric spaces and isometries. More generally, a major lesson of category theory is that changing what morphisms you're willing to consider effectively changes what mathematical objects you're studying; even if they look the same, they really aren't.