Should domestic cats have wet noses?
No, positively not unless your cat has condensation on their nose leather because they've been outside in freezing weather, or they have a URI. But under normal circumstances a cat's nose should be dry but not hard dry and therefore very slightly damp because there is always a little bit condensation and the air from the lungs is moist. To stress: there may be a bit of moisture there but to the touch the nose leather should not be 'wet' per se.
I think the problem here is that people are mixing up cats with dogs. Wet noses for dogs are normal but that does not mean that they should be wet all the time.
There are a range of conditions for a dog's nose leather. But I sense that people think that a dog's nose should be wet and the same applies to cats. Wrong, I am afraid.
Other than that, I don't understand why some people think a cat's nose should be 'wet'. That's plain wrong. It'll be wet if they have an upper respiratory infection as mentioned. Of course.
You only have to think about humans and how the end of their nose looks and feels normally. It is not wet.
Sometimes cats lick their nose as displacement behaviour i.e., distracting behaviour when they can't make up their mind. This may cause the nose to be temporarily a little bit moist but that is it.
I have lived with cats for decades and my current cat for 8 years and touch wood he has been healthy throughout those years. His nose has always been dry and at the most a little damp because of the moist air from the lungs and condensation. I don't mean bone dry and almost chapped but nicely soft but not moist or wet.
Enough said really. I could waffle on but what is the point?