What is the name, if any, of the act of equating two distinct definitions of the same word? [duplicate]

Inspired by this chat message (text is below), I have sometimes wondered if there is a word for the feeling that a person you were talking to about a particular subject was using the same words as you, but was actually talking about something else, or was leading into a very different and seemingly unrelated subject. Typically, you feel like the person is crazy by this point.

Is there a single word for this? Is there a two- or three-word phrase, maybe?

The chat message:

There should be a word for that double-take sensation you get when you realize the conversation is further down the rabbit hole than you thought. Like when you are having what you think is a normal chat about interest rates, or something, and the other person mentions that we're all controlled by lizards from space.

be at cross-purposes

at cross-purposes:

If two or more people are at cross purposes, they do not understand each other because they are talking about different subjects without realizing this: I think we've been talking at cross purposes - I meant next year, not this year.

Catch the wrong pig by the ear

the porcine reference is unlikely to be flattering but might not land you in any trouble if the person is crazy and at cross-purposes the whole time.

Talking past each other might work.