ultimately – hopefully – what you’re asking for here will be included in revamp of the notifications system. however, that’s probably not happening anytime soon. see: https://forum.inaturalist.org/t/notifications-on-inaturalist-please-share-your-ideas/374/202.
in the meantime, it’s not a search function, but this may help you view notifications related to observations (though not notifications related to non-observation things like flags, journal points, etc.): https://forum.inaturalist.org/t/is-there-an-easy-way-to-open-all-notifications/25913/35. it basically displays the same sort of information that you would see in the mobile app activity screen, with a few additional bits of info from the associated observation. when using this to get mentions on observations you’re not already following, you have to explicitly look for observations by others, as noted here: https://forum.inaturalist.org/t/api-full-text-search-of-body-tags-in-comments-identifications-and-return-observations/31170/29.