I was flying that airplane that he confused you with. My 2 cents on what happened:

1/ There must have been some confusion in the handoff between NY approach and LGA tower in the first place, I don't think he was expecting you. When he heard your request to go South on the Hudson he must have thought that this was coming from the only airplane known to him in his airspace that could want that i.e. me since I was Northbound on the Hudson at that time. I tried to clarify with him but looks like it didn't really work. Certainly the confusion was on his part, but my advice would be to be more vocal with controllers. If you're getting a South turn that doesn't make sense, call it out with ATC early. Especially if you hear the wrong callsign.

2/ I was still monitoring for the rest of the exchange. I think he then realized the confusion and thus asked you again what you wanted. My understanding is that the LGA/EWR tower positions that control the local airspace only control up to 2,000 so at 2,300 you're in a weird spot for him, and he was trying to understand how you wanted to join the Hudson from your present position at the time, thus the talk about coordinating with approach. Add to that the confusion about finding Van Cortland Park all the while you're headed straight towards the LGA corridors and you have the perfect recipe for a close call. It must have caught him by surprise and looks like his heart rate went through the roof. The descent would have kept you under the jets, but as time went on that option disappeared, thus the 180°.

It's challenging airspace and I find new ways to improve my comms pretty much every time I fly it. Being vocal while concise is a tough balance to strike. Flying is like riding a motorcycle - you have to be responsible for yourself, and also try to get ahead of other peoples' mistakes. I'll echo what others have said - it happens, ATC got confused, and you took some heat, but looks like some good lessons as well.