If you're willing to generalize from plain categories to enhanced types of categories like enriched/internal/indexed categories, then there are a lot more examples. Some famous ones are:

1. Sheaves on a site can be identified with certain categories enriched in an appropriate bicategory; see Walters, Sheaves on sites as Cauchy-complete categories.

2. Metric spaces can be identified with certain categories enriched over the poset $[0,\infty)$; see Lawvere, Metric spaces, generalized logic and closed categories.

3. Orbifolds can be identified with certain internal groupoids in smooth manifolds; see the references here.