I think the existence of limits for different types of diagrams serve as categorical invariants. Higher category theory I think is the search of more fine grained invariants but I don't know if the practitioners and developers of that program think of it that way. When comparing two categories, if all the same diagrams have limits then for all intents and purposes those categories are equivalent as far as category theory is concerned because there will be no way to distinguish between them categorically.

The Yoneda Lemma (https://ncatlab.org/nlab/show/Yoneda+lemma) says something to this effect already. If two objects in a category are isomorphic as presheaves then their objects were already isomorphic to begin with ([isomorphism of representables][1]). In other words, if two objects of a category have the same structure for their incoming arrows then they are isomorphic (and presumably there is the dual version with outgoing arrows and their structure).


  [1]: https://i.sstatic.net/57GWz.png