Off the top of my head, the closest thing I can imagine along the indicated lines is to suspend A → A on the propositional level, but even so, this would be on pragmatic, not semantic, grounds (i.e. "it's dialogically silly to infer a proposition from itself," although in terms of abstract validity, hardly any inference seems more obvious!O).
Alternatively, consider hyperintensionality, e.g. the difference between, "Sam believes that Dean is his brother," and, "Sam believes that a fictional character named 'Dean' in a TV show, is his brother," or, "Sam believes that Sam Clemens is Sam Clemens," vs., "Sam believes that Mark Twain is Sam Clemens." If we drop the axiom scheme of term substitution, then, for example, or modify it at least (to get at what you're aiming to hit upon), we might generate hyperintensionally paraconsistent sets of sentences (I suppose so, anyway).
We might also modify identity in terms of relativizing such frameworks (c.f., as the article discusses, the myriads of thin and thick equivalence relations in mathematics) or by traversing the dark anfract of transworld identity. Questions such as of predicativity also come to mind, and then the matter of object-vs.-definition circularity. You might also be interested in connexive logic:
The name ‘connexive logic’ was introduced by Storrs McCall (1963, 1964) and suggests that systems of connexive logic are motivated by some ideas about coherence or connection between the premises and the conclusions of valid inferences or between the antecedent and the succedent (consequent) of valid implications. The kind of coherence in question concerns the meaning of implication and negation (see the entries on indicative conditionals, the logic of conditionals, counterfactuals, and negation). One basic idea is that no formula provably implies or is implied by its own negation. This conception may be expressed by requiring that for every formula A,
⊬ ~A → A and ⊬ A → ~A
Note that if we defined some sentence S such that S = S + T (letting "+" be the conjunction mark), then either S = T already, or somehow T by itself doesn't equal S. I don't know how such a sentence would work, and it seems as if it might not really work very well, ultimately, so these kinds of modifications of identity formulae, in whichever intended system, must be undertaken carefully, if at all.
OIndeed, if (deductive) validity occurs when it's the case that if the premises are true, then the conclusion is true, then in case A is true (any A), then A is true. But dialogically, we object to if the premise is true as a premise, then the premise is true as a conclusion too, since this is tantamount to circular reasoning after a mere half-step through one's thoughts.