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Correction, between things that CAN be different. It doesn't make sense to put equality symbol between things that are known to be different.

Then, we'd say it is the assignment operator not the equality operator.

No, it's the equality operator. It's just that when we solve an equation we assume that equality holds and proceed until we get possible solutions or a contradiction. If we got a contradiction, then it means that our assumption was false.

For many languages == is the equality operator and = is assignment. Proof by contradiction is definitely an important technique.

I know. I was talking in context of mathematics.