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We all understand that
Given A = C, and B = C,
Then A = B.

However, A is not “the same as” B.

Example:

A is the question, “What animals have feathers and can fly?”

B is the question, “What was the primary subject of artist John James Audubon?”

C is an answer, “Birds.”

We can say both A and B = C, and this relationship is bidirectional.

“Birds” ARE “Animals with feathers and can fly”
Also,
“Animals with feathers and can fly” ARE “Birds”

however A is certainly not “the same question” as B. They are not interchangeable in most contexts.

This problem arose on this site in fact regarding the idea of “duplicate questions.” A vote to close a question was based on nothing more than the fact that it had the same answer as another question (therefore, duplicate question).

The voter’s exact argument is below:

“@VogonPoet - Fair enough. In that case, it is definitely a duplicate.
All of the other races mention in 'The Chase' are addressed in the top
answer to the duplicate question. –“

It seems an irrational logic but what is the actual logical distinction between things which have the same answer but are not “the same” - while having equal solutions?

This is a question in philosophy that deals with the metaphysics of identity. A classic problem in philosophy is the Ship of Theseus and goes back to the pre-Socratics, particularly Heraclitus and his proposition that one cannot stand in the same river twice.

In logic, one often draws a distinction between a name (symbol) and the thing it represents (referent) the relationship between the two being called the thought (reference) as per Ogden and Richard's The Meaning of Meaning though the same ideas are found in ancient Greece. And this is a useful distinction if one presumes a sharp distinction between the outside world and the inside mind like in substance dualism.

In your example, note that you are attempting to apply a property to the propositions that does not hold. A-->C, and B-->C and NB that A-/->B and B-/->A. As such, the transitive property ((A-->B, B-->C)--> (A->C)) does not apply!

This is why it is important to examine not just the symbols in an argument, but the content (of the referents) too, because meaning can be found not only in a proposition, but in the relationship between propositions. Particularly important is recognizing that equality is a bidirectional implication, and that questions and answers aren't related the same way two identical quantities with different labels are.

EDIT

NB: A=B, B=C -> A=C is defined as (A<-->B, B<-->C) --> (A<-->C) because (A-->B, B-->C)-->(A-->C) and (C-->B,B-->A)-->(C-->A) where <--> is defined as --> and <-- true over two symbols.

Non sequitur

I'll go off of the example in the comments, namely

“One dollar” = “money” : “Nickel” = “money.” Therefore, “one dollar” = “nickel.”

This is non sequitur - there's no logical reason to assume that Therefore.

Or, alternatively, this could be ambiguity fallacy as this seems to be caused by (intentional?) misapplication of the symbol "=" with different, incompatible meanings. Whatever you mean by this symbol should be properly defined - and in this case you can't use the standard definition of "=" used in math (and thus assume its properties) because that definition of "=" operates on a particular restricted set of objects (e.g. natural numbers or something like that) and doesn't apply to the objects like "one dollar" or "money".

And the thing is, that for most reasonable definitions of what you mean by "=" you'll have to choose which parts of the statement stay valid - either “One dollar” = “money” will be clearly false under that definition, or you won't be able to show that transitivity should apply for "your =" and the therefore part would be without any basis whatsoever - thus the non sequitur fallacy. You could also define "=" in a way where both these things are true, but in that case your "=" would simply mean something like "is vaguely related to the same concept" and in which case "one dollar"="nickel" would be true for that definition of "=" without any problems.

As the other answer states, you can't simply assume that Given A = C, and B = C, Then A = B - transitivity is a property that some well-defined operators have and others don't; it needs to be proven under whatever axioms and operator definitions you choose. If you stay within the bounds of some particular theory such as the math of real numbers or first order logic or whatever, then you can "piggyback" on the various properties that are proven for the symbols used in that theory; but if you make up your own meaning for these symbols (to enable you to say that “One dollar” = “money”) then you're on your own and need to start from scratch to prove what properties hold for symbols in your theory. So there's some ambiguity and vagueness caused by re-purposing "=" to mean something substantially different from the common understanding of "=". Possibly that's a redefinition fallacy, but at least for me the statement stands most clearly as a simple non sequitur.

I do believe you've missed the point of 'duplicate' here. 'Sameness' in this context is a fairly loose and utilitarian construct. Consider: if the temple priestess says she needs a statue of Zeus for entryway, and everyone in the village steps up to sculpt a statue of Zeus, well... the priestess still only needs (and will only use) one of those statues. The statues might be of varying qualities using different materials; they might depict Zeus in different actions or storylines; they might be larger or smaller... The priestess might have a difficult job choosing which one to use (and which ones to toss in the slag heap), but all these statues will be considered 'the same' for the purpose, despite their obvious differences.

We have the same situation here. Whoever voted to close your question did so not because the question was identical to some other question, but because it filled the same functional role on the site. There is a point where one must give up the idiosyncrasies of specific questions and generalize to principle. One wouldn't want a math forum where people were constantly asking for individual sums (what is 2+6? how about 7+1? what about 11-5?); why would we want a philosophy forum where people asked every possible variation on the same basic philosophical question?