Hey everyone,
So the reason why your touchIDs are not working while plugged in is that there is an interference between the finger sensor and your finger. This interference is caused by the electric field generated by the charging solenoids (conductive lines) which route from the port to the phone battery. Apple designed the 5s/6/6+/6s/6s+ device's all with the sensor directly above the charge port. The finger print sensors are designed to measure where the unique electrical conductances are passing between your finger print tracks. This is a very sensitive task when using micro electronics. Any small amount of electro magnetic interference can cause a disruption in the pattern, hence distorting your finger print reading.
Your finger print is not ignored, it is just reading incorrectly. Multiple attempts will eventually lock out your phone and require a passcode entry. Also, try establishing a touchID while your phone is plugged in. It may eventually pass if you're lucky but all of my attempts failed. This may be caused by the lack of interference protection on your particular device from a defect in manufacturing or just because it is a bad design that has not been changed for over 4 years. I can't recommend design changes on here due to copyright policies.
I had the 5s and now a 6s, and still no change in the issue. I recommend you not bother with the hassle of exchanging phones if this is your only "problem." Getting a "new" one will either replace your current problem with the same one, or worse, new one's. It is possible to hopefully get the current flowing into your device low enough such that the electric field can be reduced to a level in which the interference is negligible which is why some people are experiencing a working touchID while plugged in using the same OEM Apple lightning cable and AC-DC converter (100-240VAC to 5VDC @ 1A). It all depends on how the converter can manage the outlet Voltage and Current. At this point, is it more worth it to rewire your home, fly to other cities, or even search endlessly for a manageable wall outlet, or just live with the device you paid for. Often times they will have you reset, reformat, or return your device, but frankly those are just band aids to the real problem.