Did my rental car have a key?

There was a recall issued for my Toyota Corolla. Following instructions, I took it to a dealer in the morning and was given a free rental car to use for the day while they looked at my car.

When I got in the rental car, the engine was already running, so I did not think to look for a key. When I later parked the car, I looked for the key. I could not find any. Instead, there was a button with a label that said it should be used for turning the engine on or off. I was able to turn the engine off and later restart it. I never did find a key, and there was no buzzer that went off when I opened the door. I took special care not to lock the door.

I probably should have called the dealer to ask for an explanation. Do you have any idea of what was going on? If there really was no key, there was a danger that I could have locked myself out or that the car could have been stolen.