Presuming in this answer that the cat is healthy, the reason why a domestic cat licks his nose is because he is agitated or puzzled by something. It can also mean the cat is unsure about something. It could be something very minor. You may be inadvertently doing things which concern your cat slightly.
Happens all the time
There are a million ways that a domestic cat can temporarily feel a little bit unsure or anxious about something. For example, the owner walks into a room in a rush which disturbs the family cat. The cat looks up, unsure about what is going on for a second and licks his nose.
It is the feline equivalent of biting your nails or scratching your head while you figure out what to do next. The objective of the nose licking is not to clean the nose leather (the end of the nose) but to do something to displace the uncomfortable feeling of uncertainty. Hence it is called ‘displacement activity”.
The feline nose lick happens quickly and instinctive so you can hardly see it unless you make a point of observing displacement behavior. This almost invisible behavior applies to humans scratching their head. We hardly notice it.
Different species
Diferent animal species have their own version of displacement activities. Birds wipe their beaks on a branch. Chimps scratch their arms or chins.
Making it happen
A cat owner can make their cat show a nose licking displacement activity. You rub a coin along a comb to make a buzzing sound. Quite soon your cat will lick his lips. The sound will be an unknown causing uncertainty while the cat tries to figure it out.
The much discussed videos of cats being scared of cucumbers wold have produced a fleeing nose lick as well as a jump and scram behavior.
If I want to put my computer on my lap but can’t because my cat is nice and cosy there, he might nose lick if I try and gently remove him. He’ll complain and because he doesn’t like it but is unsure how to stop it he may produce a fleeting displacement activity.
P.S. a cat with a URI may nose lick to wash off the nose discharge. This is something completely different and obvious.
