Jewish delicatessen originated in Eastern Europe and was first brought to the USA via NY. Since NY was (and somewhat still is, although LA has caught up) the center of Jewish deli in America, it set the bar for delis across the country. As such, numerous delis around the country feature products as seen/perfected/etc. in NY such as corned beef, pastrami, etc. Some even have a NY decor/theme and/or include NY in the name of the deli. Some are started by native New Yorkers, while others are started by lovers of NY deli. There are countless delis across the country in various cities that are absolutely delicious, some of which have infused their items with their own twist, local flavors, etc. For example, Detroit is supposedly home to the best rye bread in the country due to its double-baked production method. So to directly answer your question, NY-style is essentially incorporating deli's American origin in the taste, appearance, name, etc.
That's part of the experience and perfectly expected/acceptable only in this business