most likely the hospital is worried about being sued if there is a problem with taking a taxi or uber home.
I’ve never had anyone ask for ID etc., but I’ve been escorted to/from procedures requiring sedation, and I’ve done the escorting myself.
The hospitals’ reasons are a combination of legal liability and genuine medical concern. You’re impaired and if you get into trouble, you might collapse on the floor and bleed out if nobody is nearby. There are supposedly stories where a taxi driver has taken advantage of someone under the influence.
When I had my most recent colonoscopy, my original ride was not able to come at the last minute, so I drafted my son. He hadn’t had breakfast, so when we got there I gave him some cash and told him “get some food, be back by noon”. The people at the clinic did ask if my ride was there; I said “Yes, but he went for breakfast”. Then they let me walk out to the waiting room - where he was waiting, but if he had not been, I could have walked right on out. So maybe asking for ID isn’t a bad thing 
When I had wrist surgery last spring, my husband had just broken his knee a few days before and could not drive. We had a friend drop us off, and she was coming to pick us up afterward. The people at the hospital got kind of snippy and said “Well, WE have to have her phone number to call her!” (wait - just because my husband wasn’t driving??? My completely unimpaired husband?). Then when the time came, they didn’t bother.
By contrast, a month or so after that I escorted a friend for a procedure. I couldn’t drive yet, so we took Uber both ways, and the hospital (different one) didn’t bat an eyelash at it. They cracked up when I had to sign all the papers saying what to look out for… as I was doing a barely recognizable chicken scratch with my left hand.
I know around here local taxi drivers offer rides home from the hospital. The deal is you call a cab to go to the hospital, the driver gives you his name and contact info, when it is time to leave you contact him and he comes in, identifies himself as a “friend” and takes you home. The local hospitals know about this, it is against the rules, but it solves the problem for everyone concerned and the hospital is covered. So win win.