Many city-pairs are monopolistic because of the gate ownership and industry consolidation. Minneapolis was the poster child for this for many years -- you basically had to fly Northwest to go there.
Back in the pre-Reagan era, regulation forced airlines to maintain specific prices and deliver a certain level of service (think Bell System type regulation). Deregulation brought low cost carriers like Southwest, but also allowed the traditional carriers to engage in a price war that ultimately put most of the industry out of business and brought much misery to the flying public.
They are in a perfectly competitive environment which has driven industry profits extremely low which forces them to do everything as cheaply as possible.
The airline industry is the furthest you can be from a monopoly.
[added] of course there are other industries who are more commodity like real commodities, but air lines are the text book example (literally, not figuratively) for non-monopolistic competition. While the barriers appear high, they really aren't if you have the capital. If the airlines were profitable, almost anyone could get the capital to start one. In fact, upstart airlines appear consistently even though almost none of them make any money.
I like Richard Branson's claim (paraphrased) that the way to become a millionaire is to become a billionaire and start an airline.
But that's a bit glib.
It's not actually that bad if your ambitions are not massive scale. There are any number of niche local airlines that have started with a couple of small/old leased planes even relatively recently. There are well established leasing mechanisms, and passenger planes have an extremely long expected lifetime but larger airlines are often under constant pressure to upgrade to bigger and more cost effective planes, and so there's a lot of supply.
In fact, there are websites where you can enquire about quotes for leasing full size passenger planes.
That business model for scheduled service is usually built around Federal government subsides to service infrequently visited airports. "Cape Air" is an example such airline that flies to places like Nantucket, MA and Lake Placid, NY. The airports often coincidentally happen to be where prominent congressmen live or have summer homes.
The other variant for that niche airline model is charter flights -- get a few hundred people from a city to go to Jamaica or something.
The delusion that some have that airlines represent "perfect competition" is amusing. It's an industry held up by direct subsidy, indirect subsidy via the Postal Service & military, and other bizarre compromises.
... carry people (and not just cargo) and how many actually reach the most destinations? It's pretty much the early Hollywood Studio system up in the air. Sure, you can make a movie, but good luck getting known actors in it even if you could afford to pay them.
Would you care to name the single entity that has the monopoly in question?
(I do not think that word means what you think it means.)