Does Mexico issue exit stamps at airports, or only at land borders?

There are no exit immigration formalities that would attract a stamp in Mexico for Mexicans or foreign visitors, but there are for foreign residents of Mexico leaving by air.

A visitor to Mexico fills in an FMM form when they enter Mexico, leaving part of the form with the immigration officer they see on the way in and returning the remaining part to the airline, either at the gate or at the check-in counter, to register their exit when leaving. The exit doesn't generally require seeing an immigration officer unless you have, say, lost the exit portion of the original FMM and need a new one to give to the airline.

Foreign residents of Mexico also use the FMM form to report departures and arrivals, but don't have an FMM while present in the country. Instead they fill out the FMM when they are departing, visiting immigration on the way out to get the form stamped, giving the appropriate part of the form to the departure airline and returning the remaining part to the immigration officer they see on arrival. There are immigration desks airside in the departure concourses of the airport in Mexico City, and likely other bigger airports for residents to use for this; in small airports the airline check-in counter won't give you a boarding pass until you've been to the landside immigration office. In either case the visit to immigration on the way out attracts an exit stamp in the passport.

So, while I guess it is possible a visitor might get an exit stamp if the have a reason to see immigration at departure (and most reasons to do that would be the result of problems you likely don't want to have), in the typical case those stamps will only appear in the passports of foreign residents of Mexico.

I've got a picture of a couple of versions of the stamp that I'll add when I can get to a computer that can shrink the photo.

My experience (from multiple nonstop flights from California to and from Los Cabos / SJD) is that one fills in the FMM in flight before arrival. Upon arrival, Mexican immigration stamps passports, signs off on the FMM, and returns the FMM to the traveler. Leaving Mexico, the airline collects the FMM as you pass through the airport departure gate. Mexican immigration doesn't participate in the departure, and one's passport is not stamped upon exit.

At a recent (June 2018) land crossing into Tecate, Mexico, there was no Mexican immigration check, and I proceeded right through the crossing without having my FMM seen or signed-off. Upon exit through the same border crossing, there were again no Mexican immigration exit checks, so my physical FMM remained unseen by Mexican authorities. (Because this was a land crossing, I had obtained and completed the FMM online at no charge from the Mexican government webpage, so perhaps my FMM existed in their computer system.)

Your passport may be stamped when leaving the country, or not.

The formality (which doesn't seem to be very formal) is made through the form called Forma Migratoria Múltiple (here is the electronic version of it), which is sometimes handled by the airline crew.

You can get your passport stamped at the counter of the INM (national institute of migration) that you may find at the airport, but it is not a mandatory procedure.

(My answer is based on my experience, as a mexican, but the outbound halls at the airport work the same way for mexicans as they do for foreigners).

There is no immigration check when leaving Mexico by air which is similar to the US in that regard, these are exceptions, most countries do have immigration check when leaving. The idea in the US and Mexico being that you want to check people who are coming into the country and may represent a threat, you don't really care about people who are leaving, "not my problem anymore" kind of mentality.

When leaving Mexico you have to present to the airline the FMM immigration card that you were given on the plane upon arrival but there is no computer system to check it and it's really easy to buy a fake one at copycat markets like Tepito or through the internet, they just look at it and let you through.

I personally know a guy from another Latin American country who stayed in Mexico for 10 years, of course illegally, and then left unbothered at the airport with a fake FMM card. According to him he was never bothered by Mexican immigration during his decade long stay.