What does “the New York egoscape” mean?

Any word you see that ends in -scape, Oishi-san, refers to a scene or view of something [TFD]. Thus we get amalgams like

landscape
seascape
dreamscape

and so on. The point is, these are usually spacious, sweeping vistas, the size of everything you can see with your eyes in one direction.

What Remnick is doing is describing Trump's ego as something as broad and sweeping and spacious as a seascape or landscape. In other words, he's artfully saying that Trump and his ilk (rich, upscale New Yorkers) have gigantic egos.

It's clear enough that the suffix -scape refers to a scene or vista. This form is derived from landscape and is used analogously in words like cityscape, moonscape, etc. In this case, egoscape refers to the social environment — metaphorically, the ego landscape — of New York City.

In the case of this quotation there is a clever double-meaning in that Trump's ego, with ego used pejoratively to connote an excessive sense of self-worth, manifests in the physical cityscape. (Trump is a developer, known for his extravagant skyscrapers.) So egoscape is both a metaphor for New York society as a whole, and a critique of the built environment. The sense of this second meaning is that the actual landscape of New York City is a reflection of the over-sized egos who inhabit it.

Egoscape is this context refers to the mental picture New Yorkers have of their city, themselves and their place in the world.

Every city, state and town thinks it is special, but for New Yorkers, it's New York and the ROW (Rest of the World). For proof of this, see the New Yorker cover, View of The World from 9th Avenue, shown in Wikipedia

Today's Washington Post (Thu, April 14, 2016) has an article on its front page, above the fold, titled "Trump's Slice of the Big Apple". The gist of the article is that

To these New Yorkers, the reason for backing the mogul is clear: He's one of them.