Is it normal that people in front of the bar demanding keys to my car?

While I was visiting Mykonos (a Greek island), I've rented the car for few days. I've visited many places across the island, however I've experienced unusual situation with few of them (such as Scorpios beach restaurant/bar and one another which I don't remember the name).

So I'm arriving at the huge parking with plenty of space (there was no even any private gate) then the guy came, welcoming, showing his hand and asking out of blue for the keys to my car. He (of course) confirmed he'll give it back, and giving me plastic label with some number for the confirmation.

Here are my concerns:

• They guy didn't look like he was working for the bar. Standard no-name T-Short and he couldn't show any identification, or didn't show any bar's logo. Actually nothing related to the bar name.

• He took the keys into small cupboard into small hut (also without any noticeable sign that this belong to the bar) where other keys were kept.

  There were two other people seating there who confirmed that everything is fine. This didn't looked secured, everybody can get into some random hat requesting people's car's keys.

• He said he'll drive with my car in front of the bar when I finish.

  Normally you need to register extra driver when you want to allow anybody else to drive with your car. If something would happen, I'm sure insurance won't cover any damages.

• If rented car is stolen somehow, who is responsible? What I would tell police? That I've given myself key to some random looking guy?

• There wasn't option to not give the car's key, the only option was to use another parking which belonged to another bar, and there was no any other free parking around that area.

Is it normal for such "luxury" bars in Greek islands for requesting your car keys? Is it safe and legal? In other words, what are the suggested steps in such situation?