If someone says that he lives at New York, looking at the map, is the use of the phrase "at New York" correct?
If you mean the state of New York, you would almost always say in New York, because the state of New York would generally be too large to conceive of as a point on a map. So you would be talking about within (= in) the boundaries of the state.
The rest of my answer considers New York to mean the city of New York.
It would not be wrong to use at, but most native speakers would use I live in New York (City) even when looking at a map. This is because we would not conceive of living at a "point on a map" but rather as "within" (in) the boundaries of a city.
Regarding in, at, on, native speakers use at precisely when we refer to "a place or location as a point. Like an X on a map," as this answer mentions.
Two drivers who are starting from different locations might refer to a map and decide to meet at New York.
Other examples of referring to cities with at, especially as a point on a map:
The British army surrendered to the American colonists at Yorktown.
The two armies fought at Bunker Hill. (not on Bunker Hill, which one might expect for a hill)
The world's athletes gathered at London for the Olympic games.
And native speakers in the past sometimes did use living at such-and-such place as New York.
Here's one example from over one hundred years ago:
They live at New York City. (link).
Notice the same source uses 'born at New York City', 'born at Hartford, CT', etc. But both these uses are rare nowadays.
So, the short answer to your question is that nowadays we would usually say
I live in New York
even when looking at a map, because we do not live at points on a map. We live within (= in) the boundaries of a city.
But, if you said at New York, you wouldn't be wrong." You would just be referring to New York as a point on a map.
The speaker would say
I live in New York City.
since the the city would be considered to surround them as they live there.
If they were pointing to a specific place on the map, they might say
I live at Canal (Street) and Greene (Street).
I live at 72nd (Street) and Third (Avenue).
referring to the closest intersection to where they live. This is due to the grid layout of most of the City.
Usually street and avenue are not said since the naming in unambiguous, unlike in London where
are all possibilities