What is it exactly about these flares of infrared light from Sgr A* that "confirms" it is a supermassive black hole?
We have reasonably good measurements of the mass of Sagittarius A*, thanks to measurements of the movements of stars like S0-2 over several decades. It's been well-established that the mass of the central object is $M\approx4\times10^6M_{\odot}$; this alone is fairly good evidence for a supermassive black hole, and we can constrain the size of the object with the measurements (Ghez et al. 2008). Models of the orbits of other stars have further improved these results.

The recently-published paper (Abuter et al. 2018) uses a similar technique. The motion of the flares is well-described by the rotation of a "hot spot" of gas in the inner reaches of the accretion disk, with the flares arising from magnetic recombination or some similar event. In particular, it lies near the innermost stable prograde circular orbit (ISCO) - just outside it, actually (the orbital radius can be found because we know the mass of the object and the period of the orbit, with the best-fit model of the latter being $40\pm 8$ minutes). This both constrains the object's size and even qualitatively provides support for the black hole model, as we might expect to see such events around a supermassive black hole.

Looking at those articles, I think the use of the word "confirm" is inaccurate. As far as I can tell, the term is only used by the project leader, Reinhard Genzel, in that ESO statement - and it's not a claim repeated in the paper. The team describes their results as "strong support" for the supermassive black hole model, and say that their findings are "consistent with" that theory. As scientists should be, they're cautious. The results don't definitely confirm that Sagittarius A* corresponds to a supermassive black hole; they're simply additional (excellent) evidence for it.