The pilot is supposed to read back his climb clearance, so as far as I can tell he has to readback the "climb to FL170", whether or not he wants to reply to the instruction of report passing 6000 feet with a complete readback, or a "wilco", is more or less up to him.
It is also stated that once two-way communication is established between a ground station and an aircraft, you can omit the callsigns. So if you're already communicating, and the communication hasn't been interrupted, the "passing 6000 feet" would suffice. But if the transmission is "paused" for a while, I'd be puzzled as well is someone just said "passing 6000" without a callsign?