Passport control when leaving Geneva airport

Airports don't have any specific reason to check passports and typically do not do it. In Europe, usually, you will encounter:

• Checks by ground handling personnel contracted out by the airlines. The main purpose is enforcing the airline's price discrimination/yield management operations by preventing ticket resale. They might also have other purposes depending on the destination (ensuring you have documentation to enter your country of destination to avoid fines, checking data that has to be delivered to the authorities…)
• Check by the police/border guards. Everywhere I have been, it's the police (not any private company) who operates border checks and airports have limited influence on that.

Neither of these are strictly mandatory in the Schengen area and I have flown between Stuttgart and Amsterdam without showing any form of ID to anybody. ID checks by ground handling personnel (most often when boarding) are extremely common, even within the Schengen area. Police checks are rarer but cannot be ruled out. Spots checks on arrival or, especially outside Europe, checks when reaching the airport or prior to luggage inspection (e.g. TSA in the US) are also possible.

I am not flying from Geneva very frequently these days but I don't remember undergoing a police check there since Switzerland joined the Schengen area. I once travelled from Geneva with an EU citizen who had lost their national ID and was allowed to fly with only a driving license and insurance card (which did suggest they were a resident in the flight's country of destination but did not explicitly document their citizenship).

EasyJet (one of the main operators in Geneva, accounting for 45% of passenger traffic) definitely checks ID.

As Relaxed says you can sometimes manage a flight in the Schengen area without anyone ever looking at your ID. I've certainly done this in the past.

More usually however you do run into ID checks somewhere such as for proving you're the person booked on a flight or entering a secure area (such as an airport).

A few years ago I flew from Geneva to Nice, which meant needing to use the French area of Geneva airport. This was a rather odd experience, you go through a gate rather like the ones you go through after security when you disembark a flight- you know the ones, you are now leaving the airport area, no return possible, etc... on the other side of this there was then a security person checking I had a valid ID and ticket to be there.