The analog stage is on but there will be no signal on the analog outs from the DAC if the amp the DAC is connected to is off.
Surely the analog out will carry the output signal, presented as a voltage? This will be a line-level signal of a few volts maximum. The impedance of the line-level inputs to the amplifier will be high - around 10-k Ohm up to 1-M Ohm of more. So hardly any current will flow in the interconnect whether the amplifier is switched on or not. The impedance of the interconnect will allow a tiny AC current to ebb and flow in the interconnect even if it is open-circuit at the amplifier end.