CT mentioned co-ordination- I don't know exactly what I'd classify as neural vs what I'd classify as technique (there's a lot of overlap there,) but I do think that pull-ups/chins are more technical than is immediately obvious.
An awful lot of things are going on when you do a pull-up/chin, so there's a lot of things that can limit your ability to do them. As an example, grip strength can limit you, until it gets enough better enough that something else limits you. A lot of being able to display muscular strength comes down to overcoming inhibitory effects that protect you from injury, especially injury to connective tissue. But... since you don't want to injure yourself the best way to do that is to strengthen the connective tissue. Hanging from a bar, or lifting yourself to a bar, is a great way to do this.
Another possible limiter is your ability to hold a good hollow and do the rep explosively without swinging excessively. This requires and builds a lot of "core" strength. So maybe that limits you for a bit until you get it down.
Maybe even breathing is the problem for a while. When I strated doing chins I just didn't breathe at all during my sets. Once I passed about 7 reps that stopped being an option, and I had to learn to breathe properly while staying tight.
And of course things that are clearly 100% neural, like maximally recruiting motor units and properly co-ordinating the firing of motor units between muscles can also be a limiting factor. As can muscle mass itself.
I think one of the reasons that things like pull-ups/chins improve so readily with practice is that there are so many limiting factors, and you tend to cycle through them as one improves and another takes its place as the limiting factor. Compare something like a Preacher curl, where there are just a few- you can only improve each so quickly, so you run out of room to improve very soon.
Of course the question has been raised: "Have you really gotten stronger?" I think the answer is clearly "Yes." You used to be able to do 3 chins, now you can do 10, so you are clearly stronger in that movement. On top of that, you have removed limiting factors that aren't muscular strength (and size if you want it,) allowing you to train for that with Chins where before you couldn't.
You've also improved your grip (the tendons and ligaments in the hands and forearms are important and overlooked limiting factors in many exercises and are ignored by many lifters,) improved your ability to hold your spine in rigid extension (the carryover to squats and deadlifts should be obvious, and vice-versa,) and just generally become a lot better than the you who could only do a pull-up or two.