Most likely, the Yulong DA8 II headphone amp section has some self noise. Since the sensitivity of the SRH 840 headphone is much lower than the Aonic 5, it pushes the noise below the threshold of hearing.
This means that the measurements and specs on the Yulong website are probably referenced to the DA8 II in DAC mode and not taken from the headphone out.
For the HRT HeadStreamer, it's a bit strange that the noise is related to the Windows system volume. Are you sure that Windows / foobar is giving you the correct sample rate / bit depth for the music you're listening to?
My suggestion is to find an amp that offers a very high signal to noise ratio even at very low power. A JDS Atom seems to fit the bill and it's fairly inexpensive. Try hooking it up to the Yulong in DAC mode. Hopefully you'll have a noise free system with your Aonic 5.