Galaxies do not have air. They are made up of billions of stars, but the space between them is mostly empty space.

Planets and some moons have air. All other planets in our solar system have some amount of air or air-like gas that we call an 'atmosphere'. They're not the same kind of air, though - you can only breathe the air on Earth.

We do not know of any other planet in the universe that has breathable air, but it is widely believed that they exist. We just haven't found them yet. However, we don't know how common they are.

So a galaxy is a collection of stars and planets kept together by gravity with a black hole in the middle. Since most of a galaxy is empty space, there is no air in the vast majority of a galaxy.

However, the gravitational pull of a planet is sometimes enough to keep gases on the surface which we call an atmosphere. If this atmosphere contains a gas mixture that is the same as on Earth, we would consider that this planet has air.

It is possible that there are planets in other galaxies that have air like ours, but at the moment we haven’t found any for sure.

Forgive me if I'm wrong, but I'm thinking that when you wrote 'air' you mean 'atmosphere'. If so, then most likely other galaxies have atmosphere too. It might not be the same composition and proportion as what we have here on Earth, but an atmosphere nonetheless. Unfortunately I can't tell you what kind of atmosphere it could be, as I'm not familiar with them. But hopefully someone else might be able to answer it.