The comments on the question point out that it's not really well-posed: the property "bijective" isn't defined for morphisms of an arbitrary category.
However, for maps between sets, "bijective" means "injective and surjective", and a . A common way to interpret "injective" in an arbitrary category is "monic", and a common way to interpret "surjective" in an arbitrary category is "epic". So we might interpret "bijective" as "monic and epic".
Then JHS's question becomes: is there a name for categories in which every morphism that is both monic and epic is an isomorphism? And the answer is yes: balanced.
It's not a particularly inspired choice of wordsname, nor does it seem to be a particularly important concept. But the terminology is quite old and well-established, in its own small way.
Incidentally, you don't have to interpret "injective" or and "surjective" in the ways suggested. You could, for instance, interpret "surjective" as "regular epic", and indeed that's often a sensible thing to do. But then the question becomes trivial, since any morphism that's both monic and regular epic is automatically an isomorphism.