What do you do every time you see a mirror? I spontaneously remove my phone & start clicking pictures to see how wonderful I look. 

Just so you know, there is a deeper meaning to it. So wait & read ahead. 

Source: Mental floss

Firstly, elevators are great inventions we should all be thankful for. You cannot expect anyone to climb 20 floors every day. So, it's a great relief there. 

To Make You Feel Safe.

How many times you were alone with someone in the elevator & felt unsafe. Yeah, we all do. Mirrors help you see what everyone around is doing. It also keeps you aware of any assault or robbery that may happen. 

According to the Japan Elevator Association, it is a regulation for all lifts to be outfitted with a mirror. The reason for doing so is to help wheel-chair-bound people move in and out of the elevator easily.

Source: Quora

To Reduce Anxiety And Claustrophobia

Some of us may feel scared or claustrophobic in closed spaces. A mirror helps you in such situations, as it appears that there’s more space and less crowded. It also reduces the fear of being trapped.

Source: 90' news

To Keep You Distracted

The purpose is functional, but it’s psychological rather than mechanical. Many of us get bored gazing at the elevator, or someone like me would get anxious. So, back then, elevators were installed to keep you distracted from all such emotions. 

A Quora thread also gives us more details. "In the early industrial age, many new buildings were taller than anything ever built before, and most had elevators. As buildings got taller and taller, more people began to use lifts, but as they were quite slow, people were constantly complaining."

Source: Hamariweb.com

Elevator companies were frustrated with this problem and came up with the typical problem statement: "elevators move too slow." So, they went off to design elevators that were faster and safer, but at the time, it was very expensive to do so. 

Furthermore, an engineer proposed that the problem was people think lifts move too slow. He inserted two words “people think” into the problem statement. That allowed the design team to approach the problem from a completely different perspective.

Source: story pick

Instead of concentrating on larger motors, slicker pulley designs, and such, they concentrated on the passenger in the lift.

So after long due research, they found out people had an exaggerated sense of time because they had nothing to do but stare at the wall and think about the safety of the elevator being suspended in the air and preoccupied with the fear of falling.

The next time you are in an elevator, share this with someone. So you both feel better.