Even in software development, there is nuance. Are you talking about a set of OO objects with unique OIDs? Or, for example, rows in a database with unique primary keys?
"Unique" on its own is inappropriate as it only means that the objects are all different - not that they each have a unique identifier.
"Indexed" means that they have an associated index used for identifying, and/or sorting. Sorting not happen on something other than a unique identifier so again is ambiguous. Sorting unique people by an indexed state of residence for example.
"Identified" would imply that the elements of the set each have some sort of unique identifier, but does not neccessarily imply that the entire set is unique (a query bringing back duplicates for example)
So if you want real specificity (and in software development, oh boy do you want that!) then brevity is not always best. If you want to say that the set is all identified and no duplicates exist, then say so! "This distinct (or unique) set of identified objects". And if you need to express HOW they are identified - add that too.
Shortcuts in technical docs almost always lead to imperfect documentation. You say that "The name should not necessary stress that the items are unique"? Yes, yes it should if that is what you mean! If you don't mean it - then be specific in the other way.