The Scythians were an Iranian people related to the Persians. Herodotus and other ancient Greek sources aren't very reliable when they try to explain the national origins of non-Greek peoples. Medieval Greek sources often use the classical names for people living in a certain area even if they are not related to the people who lived there in the first millennium B.C. Apparently the origin of the Khazars is open to dispute but they may be totally unrelated to the Scythians, apart from living in the same general area.
Herodotus is reliable as a source on what his contemporaries said about the origins of non-Greek peoples, though. What he says in this particular instance is consistent with what else is known about the Scythians from Near Eastern, Russian, and Indian sources. I'd agree the origin of the Khazars is open to dispute.