No, StoryTime, it's not close enough. It's not close at all. "Especially in function" is what you don't understand.
Mayor de Blasio's counterpart in Chicago would be Mayor Lightfoot: they have the same job in the other's system, they're both Mayors. Mayor de Blasio's counterpart in the Democratic party... doesn't make much sense. There's only one Mayor of New York, and it's Mayor de Blasio. He doesn't HAVE another counterpart in the Democratic Party. You could maybe say that Mayor de Blasio has a counterpart in the Republican party, and, I don't know who, I guess you could pick a prominent city mayor that the Republicans have and you could argue that THEY'RE counterparts.
But what you wrote just doesn't make sense. The three people you're talking about have COMPLETELY different jobs in the Democratic party. They do not do the same things. They're not counterparts to each other in any way shape or form. You don't know what it means.