Going along with his theory that Achashverosh was under the mistaken impression that Haman had no genocidal intentions, Malbim writes that Achashverosh’s removal of his ring, Haman’s genealogy, and even Haman’s title here of “enemy of the Jews” are all meant to describe Haman in contrast to Achashverosh, who was in no way culpable for the decree to exterminate the Jews.
The Vilna Gaon writes that Haman is simply called “enemy of the Jews” because he did not explicitly name the nation he wanted to kill. Therefore, the verse uses this appellation to clarify his intent.
According to R’ Dovid Feinstein, this phrase is meant to indicate Haman’s new role – that of solver of the Jewish Problem.
The GraMad (R’ Meshulam Dovid Soloveitchik) adds that this title is Haman’s only redeeming quality for Achashverosh.
Another reason he is called “tzorer haYehudim” (“enemy of the Jews”) is “tzorer” can also mean “binding” (see Bereishis 42:39 and Chullin 107b). Iturei Torah says that this indicates that it was Haman who bound the erstwhile “scattered and dispersed” (Esther 3:9) Jews together into a unified front at this point. Parenthetically, the reason “tzorer” can mean both enemy and binding is because, like the two definitions for the English word “rival,” one would need to be connected in a relationship with someone in order to have a deep feeling – even hate – for that person.