The names likely share an etymological root. Both Khazars and Kazakhs spoke Turkic languages, and *qaz- is a Turkic root meaning "to wander" – an obvious endonym for a nomadic people. Other than that, I don't think there's any particular connection. They were both steppe khanates, but they coalesced in different times and places: the Khazar khanate in the Pontic-Caspian steppe in the 7th–9th centuries, the Khazakh khanate in Central Asia in the 15th–19th centuries. An analogy would be the Welsh and the Vlachs. Their names share an etymology (Proto-Germanic *Walhaz, stranger) but there's no historical connection. They just happened to pick the same name out of a similar cultural-linguistic milieu.