I'm not treating it as a direct quote at all. I'm treating it as a misleading and shitty paraphrase of something that Kagan never actually wrote. And it is.
This is the misleading paraphrase: Kagan argued it may be proper to suppress speech because it is offensive to society or to the government.
The portion quoted above offers no support whatsoever for the paraphrase. Indeed, the portion you quoted above concerns the governmental interest in passing hate-crime laws -- to suppress race-based violence, not laws suppressing speech -- race-based speech. Moreover, Kagan is merely describing the Supreme Court's unanimous holding in the Mitchell case, not espousing her personal view of the matter.
Compare what Kagan wrote to this:
That's Chief Justice Rehnquist writing for a unanimous Court in Mitchell. Apparently, Chief Justice Rehnquist and the entirety of the Supreme Court in 1993 (when that case was decided), like Kagan, can be said to support the notion that the government can suppress speech because it is offensive to society or the government.