“He does not recognize them.” is also a correct translation. In the accusative case, “her” and “them” are both ‘sie’ (and formal “you” is ‘Sie’). It is only in the dative case where they differ, “her” being ‘sie’, while “them” is ‘ihnen’ (and formal “you” is ‘Ihnen’).
As a transitive verb, ‘erkennen’ takes a nominative subject and an accusative direct object. If ‘er’ is the nominative subject, ‘sie’ must be the accusative direct object. If ‘sie’ were the nominative subject, ‘er’ would have to be ‘ihn’ in the accusative case: ‘Ihn erkennt sie nicht.’ = “Him she doesn't recognize.”.
He recognizes many people he doesn't know, because he's only seen pictures of them, or encountered them often without ever getting to know them.
He sometimes doesn't recognize people he knows, because they've changed, or he only perceives them in unfamiliar or insufficient ways.
Das Erkennen = knowledge, cognition; das Kennen = knowledge (by acquaintance). In English we can say, "He doesn't know her" to mean, "He doesn't know who she is", i.e., "He doesn't recognize her", in which case "Er erkennt sie nicht" would mean for such a speaker of English, "He doesn't know her" (= he doesn't know who she is). (By the way, we would never say, "He doesn't cognize her".)
The point is that to "know" someone in English has a wide range of meaning stretching from "I know her!" or "I'm sure I know her!" with the tone of / meaning "I've seen her face somewhere" (faint recognition) to "I know her well". In other words, to know someone or something in English is rather ambiguous as a statement on its own. One would almost always need to add qualifying words or tonality to specify the kind of knowing (faint recognition to deep knowing) that one means.