A no-win situation in a debate where any response you give is considerd to support the premise?

Could fall under many different things but what I see here is some form of bias. Person 1 for what ever reason already made up their mind and nothing is going to persuade them differently.

I see this more as a communication issue and not yet a logical discussion.

Person 1 made a statement, nothing more, and (depending upon the context) you could just move on and talk about the weather. No reasoning process is  involved. (Perhaps a self-sealing argument, but I I view it as not yet at that level.) If you want to engage in a discussion, then the response of Person 2 will not lead to it.

Let me explain why I say this. In my experience, people never change their beliefs by being punched in the head with facts. Most people believe what they believe because they want to believe it. Facts are not important. Michael Shermer made this addition to Cognitive Dissonance Theory in his book, “Why People Believe Weird Things.” So, if your goal is to change another person’s belief, then you must use a different approach.

Peter Boghossian pointed out what a strategy to change a person’s belief. To be successful, he said the person whose belief you want to change must reconsider how he arrived at the belief under discussion. If your goal is to change his mind, as distinct from pontificating (which is better done in front of a mirror), then you need to get him thinking about how he arrived at the belief.  

Boghossian’s book, “How to Have Impossible Conversations” is an excellent manual on how to do this.

 I’ve come to a different conclusion and I’m having a hard time understanding where you’re coming from. I assume you must know some things about this that I don’t. Could you tell me more about where you’re coming from on that so I can understand better?      

The more ignorance you admit, the more readily your partner in the conversation will step in with an explanation to help you understand. And the more they attempt to explain, the more likely they are to realize the limits of their own knowledge and epistemological errors made along the way.

Person 1 would have constructed a self-sealing argument .

Seems to me person 1 and 2 both fail the gender identity test.  Either one could be male or both, or neither, on purely rational grounds.  Especially droll, if both person 1 and person 2 are female.  All contact by text message of course..