The use of the concepts of identity and teridentity is not to identify a thing-in-itself with itself, much less twice or thrice over — there is no need and therefore no utility in that. I can imagine Peirce asking, on Kantian principles if not entirely on Kantian premisses, Where is the manifold to be unified? The manifold that demands unification does not reside in the object but in the phenomena, that is, in the appearances that might have been appearances of different objects but that happen to be constrained by these identities to being just so many aspects, facets, parts, roles, or signs of one and the same object.
For example, notice how the various identity concepts actually functioned in the last example, where they had the opportunity to show their behavior in something like their natural habitat.
The use of the teridentity concept in the case of the “giver of a horse to a taker of it” is to say that the thing appearing with respect to its quality under an absolute term, a horse, the thing appearing with respect to its existence as the correlate of a dyadic relative, a potential possession, and the thing appearing with respect to its synthesis as the correlate of a triadic relative, a gift, are one and the same thing.