The typical notion of the sharing economy is that you're allowing more efficient usage of activity that would exist with or without your interference.
For example, car pooling is part of the sharing economy - the car was already going in that direction anyways, but now you've saved a bunch of gas by putting multiple people in the same car. Without your carpool, that driver would still be going that way - like BlaBlaCar.
Ditto if you and your neighbors got together and shared a lawnmower. You were all going to buy one anyways, but in this scheme you get to save money, and the lawn mower is sitting idle a lot less. Good of the environment, good for you, and maybe you can afford a better mower as a group. Score.
Uber isn't part of the sharing economy, since the drivers are only on the road because of you. It's a pretty typical merchant-client relationship that doesn't fit any common notion of the word "sharing", unless you consider your coffee shop to be "sharing" their coffee with you.
In Uber/Lyft's case, the supply (rides) wouldn't exist without you. The key tenet of the sharing economy is that you're piggy-backing on activity that would be happening regardless. I think it's a pretty tough case to argue that all these Ubers and Lyfts would be on the road anyways if it weren't for the people hailing them.
This is true for (most of) AirBnb as well. If you're renting out your apartment while you're away, that's a part of the sharing economy, since you'd own the space even if AirBnb didn't exist - it would just be sitting empty. If you're maintaining a room or apartment strictly for the purpose of renting out, it really isn't, and now you're looking at a much more traditional merchant-client relationship.