My question seems to be simple at first sight, but it is confusing for me. I feel that "the last chance" means the kind of chance which ends any other chances, while "a last chance" is like one of many last chances. I would go for "the last chance," but the problem is that I can see quite a few native-speakers' opinions on the internet that "a last chance" is the right choice while "the last chance" is wrong, yet, it is used in a sentence like "I give you the/a last chance". Is it "a" or "the"?
"Last chance" pretty much implies that there is only one. You can't have two "last" chances, or one of them wouldn't be last. So normally you would say "the last chance".
I suppose if you were talking about your last chance to do X and also about your last chance to do Y, or about Al's last chance and Bob's last chance, then you might refer to one of them as "a last chance". But I think that would be rare.
Perhaps you could give an example where someone said that "a last chance" is right and "the last chance" is wrong.
I claim the problem here is not uniqueness; it's that the sentence does not mean what it says.
Consider "please give me a second chance." I don't expect any native speakers to have any objections to that one, even though obviously you only get one second chance.
The problem with "Won't you give me a last chance?" is the simple fact that, however many chances you have had so far, you already had a last chance. (It was whatever you just failed at.) The question isn't really asking for a last chance; it's asking for another chance.