I'm working my third apt manager gig. The first one lasted about three years, the second one four years, and I'm in the middle of the first year of my third one.
I don't know if you can really generalize about this since every property, every landlord, and every property management agency is different. With each, you have good ones and bad ones, competent ones and incompetent ones, pleasant ones and not-so-pleasant ones.
I don't get 'weary' of seeing the same tenants everyday. If anything, it gets easier over time having the same tenants because I get more used to what they want (good and bad) and I get better at interacting with them and helping solve their problems.
But mostly, I don't get 'weary' of them because I will only work for properties that have decent clientele and a decent landlord. With a good property, you don't have to explain the landlord being a slumlord or make excuses for him/her. With good clientele, they don't whine about stupid stuff or make problems for me (behavior, late rent, etc.).
With my property/tenants, I usually never see them and they rarely come to see me. Because the property is well maintained and the landlord does the upgrades and improvements he should, my tenants are generally pretty happy and know that this is a good deal for the rent they pay. And because we screen applicants well and won't take anyone just because we want the rent, we get good tenants, too.