you could create a traditional project to hold observations of interest, although not everyone allows other folks to add their observations to traditional projects.
if you want to search through your favorites, this describes a couple of ways to bring favorited observations into the Expore (or Identify) web screen: https://forum.inaturalist.org/t/is-there-a-way-to-bring-your-favorites-into-the-explore-tab/52605. (the bottom of this post describes how i would implement this sort of functionality in the system: https://forum.inaturalist.org/t/add-search-feature-in-favorites/2502/19 .)