I am ending my rant about Uber, though I still stand by what I have said. I am sure that using Uber that way occurs in Canada as well. However, I still believe it is not good English despite being a common phrase. This is especially true as Uber branches into other forms of transportation... enough said.
As far as ""They booked a taxi car", goes. Certainly this would be never said. Taxi is not a company name. It means a car for rent with a driver. So you are essentially saying "car" twice"
Is it correct to say "I booked a shared car through Uber" or "I booked an Uber shared car"?
I am getting confused here. They are saying that "To book an Uber" is correct but you are saying "To book an Uber car" is correct.
"I booked a car through Uber.
This is the better sentence in my opinion whether it is written or spoken. "To book an Uber" may be used in common speech. Many things we say in common speech are not good English or grammatical.
If you are learning English, in my opinion
you should learn proper English first. The slang and the lazy way of talking will follow eventually as a natural result of talking to others. Other teachers' views may differ and you are seeing such right now sadly.
It may vary with the country, but in North America you don't order a taxi based on how many people are going to ride in it. This is unless you want a ride for over four people. If this is the case you would order a larger vehicle such as a mini-van if the company has one available.
So normally you would not need the word "shared" when booking a taxi.
I am sharing a taxi every day with other passenger.
(note your spelling error for "passenger")
Yes, I can imagine some sort of sharing arrangement if one was traveling by taxi every day. In this case you are telling someone about your arrangement, not ordering the taxi this way.