I believe your argument strategy can be understood without thinking about analogies at all, assuming your example conveys well your general idea. I will show why and my answers will rely on that.
From your example, the person claimed the strong statement of the form "the only thing that satisfies property X is Y" (namely that the only marriage that is valid is christian marriage)¹. You were simply testing the person if they really believe in that strong statement, by checking other things beyond Y that perhaps could also have property X, in which case the person would be forced to admit that their original premise is flawed, forcing them to look for another argument. Looks OK to me.
Just don't forget that this is a strategy to counter the person's argument, but it doesn't mean you're right either. They can find another argument, perhaps by fixing their premise, replacing it with a weaker claim.
Answering your specific questions:
- Is my reasoning wrong?
No. It's valid. You took the premise of "your opponent", which is a sentence of the form "For every X, P(X)", and tried to prove it is false by checking some specific X and seeing if "your opponent" backs down from this strong claim.
- If it is not, is there a name for this kind of reasoning?
I'm not sure, but you could call it "disproving the premise by counterexample".
- Is it a fallacy to respond "the cases are not the same" but without pointing out what the differences are that make the argument not applicable in the other context?
Recall that "the cases are not the same" is just a statement, and statements can't be fallacies (only arguments can). But clearly this sentence is being mentioned with an implicit reasoning that "therefore I should not have to agree with that". And now, it's still not exactly a fallacy in my opinion, but definitely a bad argument for several reasons:
a. "the cases are not the same" is a trivially true statement if taken at face value: as you said, obviously they are not the same, strictly speaking. But that they are not the same is irrelevant, given that they basically claimed that "every marriage except a christian one is invalid", so it should still apply to that as well.
b. Considering the above, most likely the person meant something else, implicitly, such as "when I said 'the only one', I meant 'the only one among the ones in a certain group'" and "they are not the same" is implicitly saying that your suggested counterexample is (conveniently) not in that group.
Too many implicit ideas and definitions (what exactly is that certain group?)
In short, "they are not the same" is irrelevant if taken at face value and too imprecise, full of implicit ideas, to be convincing. I'd suggest replying "what do you mean?" until they either change their premise to exclude your suggested counterexample (at which point you can follow up with 'why are you excluding that?' or trying another counterexample) or they find a way to convert that sentence in an explicit argument.
¹ I used the term "strong statement" because it is a statement that claim something about many things (namely every type of marriage, claiming that it is either christian or invalid).