Battery Stats Are Vastly Incorrect
I would start by holding all three physical buttons for a full 3 minutes (phone switched off) to fix battery logic problem. This resets the battery logic on some phones and how the battery talks to the software to let it know how much power it has and other parameters. Service center may be able to help you with exact procedure for your phone (please share their recommendation so that can help others)

Few possibilities if above didn't help:


Battery houses the chip that communicates state of battery to device including charge levels, temperature etc. That could be bad, though your phone is new, being a defective piece. Can be checked by replacing battery
Fuel Gauge drivers in OS corrupted. After backing up data,reflash stock to rule out this possibility. Also eliminates rogue app possibility, though app mischief not likely on your case
Problem with hardware (Fuel Gauge chip). Unlikely but possible


Edit

Further to our discussion, saw this on Nexus forum, which sends to have worked for most, though none reported your problem https://productforums.google.com/forum/m/#!topic/nexus/D7er7nMbzHI


  1.Power off your phone.
  2"Press and hold the Power and Volume Down buttons. This will put the phone in fastboot.
  3.Use the Volume buttons to scroll through the menu until you see "Bootloader logs"
  4.Press and hold the power key for a long time (7+ seconds) until the phone reboots.


3 button procedure described in my answer as first option is more elaborately covered here http://forums.androidcentral.com/verizon-htc-one/394469-battery-charger-reset-procedure-htc-one-m7-m8.html

Let me know which works...