You might say pretty much exactly the same about cooking, but I still think it's fair to hold professional food serving businesses to a different standard than me inviting a friend over for dinner.
Right, the way it works is that if you run a restaurant you need to be licensed, and the city or county will become cross with you if you feed many people anywhere but inside of a restaurant. So if you feed enough of your friends at your house on a regular basis, the city is going to want you to get inspected and licensed. The premise is public safety.
But wait, taxi drivers are at more risk from their fares than the other way around! Arguably, it's passengers who should have to pass a background check, and have their identity logged. If we're talking about vehicle safety, shouldn't all vehicles be getting inspected, especially if they have many miles put on them? Shouldn't all drivers be responsible drivers who are familiar with the rules of the road?
As for insurance, the rates reflect the risk and letting commercially operated cars pool with your average commuter unfairly shoves their risk over on us.
Uber provides additional insurance while carrying a passenger, and the driver's ordinary insurance premiums are already based on mileage, so if they put more miles on their car in between passengers that's also already accounted for by their insurance. Their risk is already being accounted for without you having to pay for it.
This isn't a phone book, Uber is taking a cut for every ride.
What does that have to do with anything?
Imagine a P2P program with a central server that charged you to pair up, if you want to download game of thrones season one that'll be $0.50 of which we'll take $0.10 and the uploader $0.40. Oh and the peers are legally responsible for whether the files are legal, we're only a matchmaker. How long do you think they'd be in business?
Not very long, because they would be knowingly enabling illegal activity. Nobody is arguing about how legal Uber is. The argument is about how legal Uber should be. It's notable that if any of us tried this as an individual they would just take our car and put us in jail, but as a corporation Uber is able to mount a meaningful defense, and actually go to court and make their own arguments. Corporations are the only entities with meaningful legal rights any more, because they can afford them.
But more to the point, in your supposedly congruent example they are helping to resell someone else's "property". That example is only applicable here if you assume that the monopoly that the entrenched taxi operators and services have purchased legitimately best serves the public interest, and many of us do not.