As far as I'm aware, no. In the West, logic is strongly rooted in the Law of the Excluded Middle, Law of Identity, and the Law of Non-Contradiction. It is possible to have a system of logic where one can have a statement (i.e. S :='P=Q') and accept that it AND it's negation (S* := 'P≠Q') are both true. This is called a dialetheia, but strictly speaking this isn't the same as a relation between identity and the negation of identity, although it seems they're inter-related. Also related are multi-valued logics and concepts like paraconsistent logics which deal with the explosion principle and fuzzy logic provides a quantitative formalism.
From your example:
A drop of water is the ocean. (W=O, They are the same.)
A drop of water is not the ocean. (W≠O, They are different.)
Note that sameness and difference are negations, and so the first statement is really the negation of the second. To a Westerner, this idea seems more prominent in the East than the West, not only in Indian philosophy but also East Asian philosophies like Taoism and Zen, where the latter routinely uses such contradiction in koans. Looking around, I can't find a formal logic equivalent.
The non-difference relation seems to be dealt with in informal logic by explaining the contradiction or characterizing one of the propositions as a 'literal truth' and the other as a figurative one such as a 'metaphor'. To wit:
A drop of water is the ocean, and yet not.
Were this a line of poetry, one might analyze the text by explaining that a drop of water is similar to the ocean because both are wet, thus an analogy is at play. Yet, it is a literal truth that a drop of water is not the ocean. To call it the ocean is just a metaphor.
In order to deal with the nature of the contradiction, let's look at the SEP:
A dialetheia is a sentence, A, such that both it and its negation, ¬A, are true. If falsity is assumed to be the truth of negation, a dialetheia is a sentence which is both true and false. Such a sentence is, or has, what is called a truth value glut, in distinction to a gap, a sentence that is neither true nor false. (We shall talk of sentences throughout this entry; but one could run the definition in terms of propositions, statements, or whatever one takes as one’s favourite truth-bearer: this would make little difference in the context.)
Lastly, with fuzzy logic, it is possible to have fuzzy membership so that relationships can have a three-valued membership based on a set of membership functions like min(x), 1-x, max(x). According to WP, there are fuzzy first-order logics that use general and existential quantification. You probably can find more information about these logics on one of the SE Math sites.