What's the name of this fallacy?

I think the question is kind of vague. Two words, as Rationalissimus of the Elenchus stated, can mean the same thing. But I think you need to expand your question further. 

It is intentionally vague because it is a fallacy of "ambiguous assertion".  

I’m not sure why, but on my phone it shows that the Answer I posted is a replica of R.K. Borill’s Answer. Here is my real Answer, reposted as a comment to myself. Hopefully, this works. 

Hi, Jacob!

What a great question. Rationalissimus of the Elenchus has an interesting perspective on what it might be that person 2 (P2) is banking on. But it seems to me that P2 is not trying to make a point about the meanings of phrases. X and Y do not each represent a proposition, after all. Rather, X and Y each refer to an object out in the world. P2’s point is that picking out an object, call it X for simplicity, and picking out an object, call it Y for simplicity, entails that two distinct objects have been picked out. 

P2 is mistaken, but his mistake is very understandable. The act of picking out an object is an act that can be directed towards the same object. Let me explain this by using pointing as a familiar way of picking out objects. I may point to Tom Hanks and then put my finger down. Next, I may point to Tom Hanks, again. I have pointed twice, but to the same thing. Two acts of picking out an object have been performed, but both times they were directed at the same object. This homely example shows that it is logically coherent to say that there is some x and there is some y, and x is identical to y.  

We know P2’s conclusion is false, but where exactly is the flaw in his argument for this conclusion? He argued “X and Y are different, if they were the same we wouldn't have to refer to them as X and Y; only X or Y (basically only one thing would exist and not the other).” 

The mistake is made when he says that “we wouldn't have to refer to them as X and Y.” He is misunderstanding the role that the variables “x” and “y” are playing. The variables “x” and “y” do each refer to an object, but what’s significant about using different variables each time is that it indicates that two different acts of pointing out have been performed. By saying that there is some x, we perform one act of picking out an object. By saying there is some y, we indicate that a second act of picking out an object has been performed. The variables “x” and “y” help keep track of two distinct acts of picking out an object. But as I showed earlier, those two different acts can be directed at the same object. 

Thank you, Jacob

From, Kaiden