Cat dander and hair is a very real thing. I doubt you vacuum your furniture, bed, floors, and whatever other caries your cat finds to brush against every day. No matter how well you brush the cat, the fur is fucking everywhere. It's on your clothes, it's in your car, and it literally floats in the household air.
There's nothing you can do to escape this reality, if you own a cat you pretty much accept that your shit is covered in cat hair and you have a permanently vaguely stuffy nose.
Not to mention the fact that no matter how well your cat licks itself, it can't lick away fecal bacteria stuck on the bottoms of its paws and near its butt hair, which it pads around your entire house with.
Don't worry, I absolutely believe dogs are as messy if not worse.