Here is some follow-up information:
I tried different chargers, moved my phone away from other electrical devices when charging overnight, etc. Every iPhone I have ever had has always displayed the auto-on functionality when unplugging after an overnight charge. After weeks of not turning on, three days ago my iPhone started to auto-on again! I did not do anything I'm aware of. I had NOT updated to iOS 9.1 yet when the functionality returned.
But my wife's iPhone 6s has lost her auto-on functionality in the past couple days! I can't say it started after I updated her iPhone to iOS 9.1. But I do know her phone did auto-on until relatively recently. In addition her phone does not display the large battery level icon you normally see when plugging IN the charger.
I had a genius appointment about this. The Genius was stumped and there were no records of such problems in Apple's data banks. She said she thought it might be a software glitch but she wasn't sure. I have read of others having this problem over the years. No one has posted a follow-up with an answer explaining the problem.
I am guessing this issue occurs more often than it is written about as many people probably don't take note of what their iPhone does when plugging in and unplugging the charger. I have asked about 5 people if they have ever noticed their iPhone doing this. All five said they didn't notice and really didn't look to notice.
I am going to do a hard reboot of my wife's iPhone before she plugs it in for the night. If that doesn't help I may restore her phone from backup to see if that helps as that will reinstall the software and firmware on her phone. If any of this works, I'll repost.
Update: I did a hard reboot of my wife's iPhone 6s. (Hold down the home and on/off buttons simultaneously until the screen went black and continued to press until the Apple Logo appeared on the screen. Then release the home and on/off buttons.) Boom! The next morning the screen did auto-on when I pulled out the charging plug from the phone. It also worked the next morning. Maybe a coincidence. Maybe "restarting" the phone via a hard reboot clears out some nastiness from the phone. I will add that I am almost sure I tried this with my iPhone 6s Plus and it did not work. Why my iPhone 6s Plus started to display this auto-on functionality again after weeks on not doing so remains, in my opinion, a mystery.