Unique and interesting question. It's hard to give a firm "why," but we can discuss the ways that the ear and other "holes" differ.
Mucous membranes line every tract into/out of the human body. Technically the ear is not a point of entry, as the tympanic membrane makes the external auditory canal a sort of blind pouch. The skin lining the ear is keratinzed skin. Most skin has keratin and is difficult to permeate. But mucous membranes are moist, thin, usually highly vascular, and generally easier to permeate. Disease often takes hold or gains entry at mucous membranes (respiratory infections, GI infections, etc). That is why mucus is so critical as it is protective and acts as a barrier and flushing-system.
Mucus requires some form of circulation, as it is a fluid. Once produced, usually at least some of the fluid is reabsorbed, and the remaining material is passed along until it exits the body. In the lungs, cilia bring it up to the throat and it is swallowed into the GI tract. In the nose, it drains down the back of the throat and is swallowed. In the eye, the fluid (which has mucus components to it) drains into a small tract that runs into the nasal passageway, which drains along with the other nasal mucus. Obviously saliva, which also has mucus components, is swallowed (2 liters a day of it, actually). In the genito-urinary tract, there is both reabsorption of some of the fluid components, but mainly flushing outwards when urinating. The GI tract reabsorbs almost all water components and the remnants pass with stool.
Mucus is produced by mucus glands. Cerumen (ear wax) is produced by sebaceous (oil) glands and apocrine (sweat) glands, but is primarily composed of shed skin cells from the ear canal. Sebaceous and apocrine glands are all over most of our exposed skin (except palms etc) so it stands to reason that they would be present in our ears as well.
Cerumen exits the ear several ways: all skin on the body sheds the top layers as new skin grows from underneath, and in the ear, the skin cells also migrate with a directionality from the tympanic membrane outwards. Head movements (like chewing) help dislodge things stuck to the walls.
I'll put in my plug here about NOT using cotton swabs (aka Q-tips) to clear your ears. It generally just pushes the wax deeper, except for the tiny bit you manage to get out. I've cleaned out tons of wax-impacted ear canals from people who use Qtips on themselves or their kids!!
If the ear were to use a mucous membrane system, the lower viscosity fluid would have to go somewhere. We would be weeping mucus from our ears unless there were a drainage system into the sinus passageways. The body conserves liquid, so we couldn't afford the constant fluid loss otherwise. The middle ear has a canal - the eustachean tube - which leads into the nasal sinus (although primarily to equalize pressure). But the external ear does not, so it's probably a good thing that we don't have mucous membranes in our ears.