In the chorus, Billie is using the statement to make fun of the person about whom she is singing, implying he believes he is ‘the man,’ on the basis of this philosophical idea: if you think it, then it can have some level of certainty.
However, as the song progresses, Billie clearly does not feel this man is what he believes himself to be, and in the subsequent verses seemingly points out how he is faking a relationship with her.
She tells the man to take her “pretty name out of your mouth,” and claims his world is an “ideal,” meaning his view of their relationship is an illusion.
Later, she specifically asks him not to speak about it so the press will not write about their so-called relationship, saying when the press “say your name, I just act confused.”
Billie then, in the bridge, laughs as she pretends she does not know who he is, asking whether he can remind her of his name.