Peirce’s categories are best viewed as categories of relations. To a first approximation, firstness, secondness, thirdness are simply what all monadic, dyadic, triadic relations, respectively, have in common. At a second approximation, we may take up the issues of generic versus degenerate cases of 1-, 2-, 3-adicity, but it is critical to address the first approximation first before attempting to deal with the second.
In that light, thirdness is a global property of the whole triadic relation under consideration and it is a category error to attribute thirdness to any one relational domain or role, much less any of the elements that belong to the domains or fill the roles of the triadic relation.
As it happens, we often approach a complex relation by picking one of its elements, that is, a single tuple as exemplary of the whole set of tuples that make up the relation, and then we take up the components of that tuple in one convenient order or another. That method lends itself to the impression that -ness abides in the -th component we happen to take up, but that impression begs the question of whether that order is a property of the relation itself or merely an artifact of our choice.
Failing to examine that question puts us at risk for a type of error that I’ve previously dubbed the Fallacy Of Misplaced Abstraction (FOMA). As I see it, there is a lot of that going on in the present discussion, arising from a tendency to assign Peircean categories to everything in sight, despite the fact that Peirce’s categories apply only to certain levels of structure.