I've never heard of an ANR being powered without it's own batteries on Boeing or Airbus. Is this possible?
I believe you are correct in a general sense - both Boeing and Airbus aircraft provide power down the microphone lines for the microphone itself - I believe this is 18V DC. This has been standard for, well basically forever.
But this is not the same power needed for the ANR feature - I am most familiar with the Bose Aviation X, which includes wiring details to allow additional power for the ANR circuit and involves a different headset connector and wiring behind the bulkhead connector to source the extra power.
It *may* be possible that some headset manufacturers have figured out a way to use the mic power voltage to power the ANR, but I have not seen this personally.
The Boeing 738 uses what I call the conventional twin plug GA arrangement of two jacks for the headset, while Airbus uses a 4-pin XLR. Adaptors are available and simply connect the pins from the 4-pin XLR to the two connectors of the GA arrangement, so any voltages applied to the mic line are passed directly. However, as with most things in life, any time something is more complicated than it needs to be, provides a point of failure - having the headset re-wired should be relatively cheap and take no more than 30 minutes or so.
Regards, GY