are two topics fully irrelevant to each other if and only if they share no possible variables, and yet then do all topics share at least implicit variables pertaining to fundamental categories of nature or language?
Yes and qualified no.
This is a question I have been reflecting on for a long time. I don't believe there to be a pat canonical response that ends in an SEP article, so I'll do my best to respond referencing some canonically adjacent topics.
One of the first questions I answered on this site revolved around a standardized test question which hinged on two capacities, the first was to reduce the natural language of the question to a FOL-compliant syntax that produced two facts, as I recall, in the form of:
Fx
G(y&z)
Or something close. What stuck out at the time was that it was not possible for me to further abstract away from the natural language to show a clear syntactic relation between the two FOL facts. Instead, I simply remember relying on the intuitional capacity to assess the overall relevance between to two facts. I hadn't yet recognized the existence of material logic, as it is often called, in the relations that inhere to natural language grammars above and beyond the stock formalisms like PL, FOL, and modal logic. A healthy does of natural language processing has cured me of that blindspot.
Four years later, an introduction to Montague Grammar and other formalisms, and an understanding of contemporary views of AI-centric description logics, and what I've come to believe is that relevance of propositions is normative rooted in teleological aspirations of the use of grammars and lexicons. What is relevant, in traditional natural language, often itself requires justification, and therefore there is no single justificational logic to assure relevancy. This is argued tentatively by Toulmin in his Uses of Argument when he talks about domain-specificity of argumentation and figures prominently in the notion of 'warrant' according to his model. This is also why relevancy plays prominently in the criteria applied to identifying fallacy in informal reasoning in Damer's Attacking Faulty Reasoning. John Searle often just waves his hand and calls it The Background.
Without going into detail about the differences between grammars and lexicons, suffice it to say that lexicons can be seen as terminal logical structures that occur in natural language grammars that translate observation to language. I'm going to appeal to Ray Jackendoff's Foundations of Language because it is my current paradigm for understanding the interplay between brain, meaning, grammar, and evolutionary epistemology in the broad sense that Popper suggested.
Thanks to Frege, we have some terms to work with. On the one hand, we have the capacity for sense and reference and are afforded the principle of compositionality to see how natural language grammars are generative procedures embodied by the language centers of the brain. This is the foundations of understanding syntaxes as logical structures per se. We can thank Wittgenstein, for providing us the approach to understanding semantic grounding lying with the objectives of the speaker in the language game. Therefore, when it comes to variables relating to variables, in the abstract, the mechanism for reference knows no bounds. Any variable (linguistic structure) can be related to any other in the same way in a compiler we can create references between any variable with any data value (using pointers as bindings) or create a pointer from one variable to another variable (using pointers as references).
But, usually we do not take full advantage of that. Instead, in computer compilers we apply lexical scope. In the language of logic, we declare domains of discourse where one set of variables are not associated with another, and any attempt to create grammatical constructions in the programming language embodied by the compiler trigger a compile-time error since the two variables are not allowed to know about each other.
So we see an isomorphism in natural language, where we can create language between any two terms as we see fit. This is Chomsky's example of colorless, green ideas sleeping furiously. But despite that we create relationships among these linguistic tokens by using them in our grammar (let us understand that it is always a metaphysical act to define lexicon terminology such as colorless-ness, green-ness, idea-ness), it often is meaningless to do so. There is no clear semantic grounding of a colorless idea. Therefore, according to our ambitions of creating and using a domain of discourse about 'idea', and one about 'colors', we have fundamentally disallowed knowledge-that which creates a relationship between the variables.
So, there is no inherent constraint on associating any two linguistic variables (tokens) in the act of reference, but at the same time by convention (the rules of the language game) we disallow the relation, all the while reserving the right to implement a relation (establishing a new convention by stipulation).
When you ask, do all topics share at least implicit variables, the answer is yes and no. Yes in that we are free to generate an utterance that attempts to share implicit variables, and no in that we have to do it according to the grammatical convention for it to be meaningful and useful upon pain of idiosyncratic construction. Again, in our example, when we invoke the phrase 'colorless idea', we have connected two metaphysical variables, but have done so by violating conventional definitions of colorless and idea since they habit disparate discourse. Yet, here's the rub. We can simply invoke a meta-domain of discourse to connect ANY two variables meaningful.
Consider now this metalinguistic construction: "'Colorless" and "idea" violate conventional semantics because colorless is a modifier that is inapplicable to idea since metaphysically, 'colorless' applies to a physical domain of discourse and 'idea' applies to a mental domain of discourse." Here we now have created a meaningful (grammatical and sensible) construction using two variables which would be disallowed in the language itself!
This then brings us to your titular question:
Are there any pairs of things that are fully irrelevant to each other?
No, because any set of things understood as linguistic variables have some form of relation in the domain of discourse of language itself as exemplified by the use of linguistic metalanguage to talk about variables from any seemingly disparate domains of discourse.
