I suspect that, in universe, it's a combination of factors.
Believe it or not, being stabbed by something very sharp doesn't hurt much, not initially. It's in between a hard pinch and maybe a thump. That's when it's from the outside. Well, that's my experience. It's dull things that hurt, even when they're shallow ( maybe especially when tbh).
So, you've got claws that are extremely sharp, sharp enough to shave with at least (which isn't secondly exactly the sharpest kind of sharp for blades, but close enough). They're usually drawn having a relatively acute taper as well. Not needle like, but akin to some of the modified saber, spear, or tanto point knives that are on the market.
They're also shown as coming out fast, as in less than a blink. As anyone that's had a lot of injections can tell you, a fast one is less painful than a slow one. Having been stabbed, I can say that applies to weapons as well. A nice, fast stab with an ice pick was barely noticeable at the time, but someone trying to drive a switchblade into my arm as I was handling them was maybe a 5/10 level of acute pain.
So, chances are that it would be a split second of sharp pain, followed by rapid healing that would end the pain almost as fast. Having had to clip pieces of skin off of myself post-injury, I suspect it would be similar to that, only way faster. The pain of that is gone in less time than it takes to get clippers or scissors in place to make the cut.
Anyway, I can also attest that one gets acclimated to minor acute pain, to the point it's unnoticeable most of the time. It's also very possible to prepare one's mind for minor pain to the point it doesn't hurt, it's just another sensation like being poked or rubbing against something.
Wolvie has been shown to have trained in meditation, and to have absurd levels of pain tolerance.
So, I suspect that he would normally not even notice it beyond what a particularly sharp mosquito bite would cause. Certainly far less than when, in 1998, The Undertaker threw Mankind off Hell In A Cell, and plummeted 16 ft through an announcer’s table
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