No they aren't. Public utilities generally give you only one choice of provider, which is why they need to be regulated because of their monopoly status.
When you fly, you usually have a choice between lots of airlines. So there's nothing "public utility" about it whatsoever.
Airports, on the other hand, are considered public infrastructure. There are also sometimes routes that are only served by one airline, which are sometimes regulated accordingly. But that isn't the general case, nor should it be.
Taxi companies, moving companies, and rideshare are all considered utilities and there are generally multiple choices of provider. Being a monopoly is not a requirement to be a utility.
I think the governments only role is to guarantee the planes don’t fall out of the sky or crash into each other, and then the airlines can price compete.