Using the surname of a mathematician as a metonym for his famous conjecture seems as counterintuitive as saying Christian fundamentalists don't believe in the Darwin or that some new discovery in astronomical physics substantiates the Einstein.
No one seems to object, however, if someone admires “the Dior she wore to the Golden Globes” or wonders how many Rembrandts might have been destroyed in World War II. A proper name as a metonym for something created by that person, then, is not in itself unusual, but seems so in this context.
In mathematics jargon — which I assume these authors are employing to imply their membership in this discourse community — the name Poincaré, with or without the accent aigu, is being used by some writers as a technical term for Poincaré’s conjecture and associated terms such as map, plot, or surface, as Peter Shor has so kindly pointed out. Once the writer establishes which particular one is meant, as the New Yorker article does with conjecture several times, then it becomes the Poincaré just like the Dior at the Golden Globes.
A simple Google search for “the Poincaré is” yielded so many hits for a variety of omitted nouns I didn’t bother to count them or search using another verb. Such a search cannot, of course, determine how many mathematicians use such a construction as opposed to those who don't or who object to it. All we can know from the posted question is two writers for the New Yorker and some dude grousing about it on the internet.
Ultimately, this is a prescriptive-descriptive conflict to be decided by the discourse community that uses the construction.
A few samples show both the variety of contexts and the almost pronomial use of Poincaré:
The Poincare is a central question in topology, the study of the geometrical properties of objects that do not change when they are stretched, distorted or shrunk. [Poincaré conjecture] —BBC news report.
Although the Poincaré is useful visual pattern for HRV, it has limitations. [Poincaré plot] — International Journal of Medical, Health, Biomedical, Bioengineering and Pharmaceutical Engineering Vol:9, No:9, 2015.
If the Poincaré is a finite set of points, then the corresponding system motion is periodic motion state. [Poincaré map] — Kehui Sun, Chaotic Secure Communication, DeGruyter 2015.
The Poincare is just what he said: the group of symmetries of flat spacetime. [Poincaré group] —Physics Forum post to question “What is a Poincaré group?”
In the proof of (ii) we use the Poincaré map and degree theory. The following lemma guarantees that the Poincaré is well defined. [Poincaré map] — Journal of Mathematical Analysis and Applications 185(1994), 480.
But the implication generally does not go the other way, and since the Poincaré is true there was no special number. [Poincaré conjecture] — "Gödel’s Lost Letter and P=NP,” Blog by Prof. Dick Lipton (Georgia Tech) and Prof. Ken Regan (SUNY Buffalo).
In this paper we discuss the natural candidate for the one dimensional free Poincaré inequality... As in the classical case the Poincaré is implied by the others. — Trans. Amer. Math. Soc. 365 (2013), 4811-4849.
From peer-reviewed journals and monographs to blogs and forum posts, some writers, having specified unambiguously which Poincaré x is under discussion, then employ the Poincaré for subsequent mentions. While this metonymic usage may be the mathematician’s version of the split infinitive or sentence adverb, the variety of genre and the status of the authors within the discourse group — seriously, are you going to tell a tenured professor at Georgia Tech how to write? — suggest that, regardless of one's personal opinion, the usage will not likely disappear any time soon.