This begs the question: what _is_ angular momentum?
How does it gets transferred from the swirling gas to whatever degenerate state of matter is inside the neutron star? What kind of interaction is that?
Almost nothing falls straight into a star/neutron star/black hole. For it to fall straight in, it would have to be moving exactly towards the location of the s/ns/bh, which is very unlikely (and won't happen for lots of matter at the same time).
Generally, matter that's near a large mass is circling in a disc; if it isn't, then the particles will collide with each other until they've averaged out their motion. So, relative to that large mass, there will be a whole lot of angular momentum in that matter. The angular momentum doesn't go away as it approaches the mass; if it manages to hit the mass then the angular momentum will be added to the mass.
Fun fact: it's hard for us to send an object to collide with our Sun, because you have to cancel out all the angular momentum that everything on Earth carries.
One nice perspective on what angular momentum _is_:
It's a consequence of the laws of physics being the same for all directions.
Noether's theorem can be paraphrased as:
If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.
If the laws of physics are the same even if your coordinate system is rotated that means there is a continous symmetry for that. When The conserved property is written out is is angular momentum.
I think it's an interesting question, it got me thinking a little bit (ianap). If you break down thinks at the microscopic level, angular momentum seems to not be a thing in itself, but rather a combination of momentum and some force that keeps things from going away from a center, like gravity in the case of interstellar gas, or chemical bonds in the case of a solid object. So what gets transferred in the end is momentum of all the gas particles that get assimilated in the neutron star. That begs the question, what is momentum? I don't really know, I guess it's a fundamental property of matter that can't get broken down further.
Im not completely confident, given my knowledge isn’t by an understanding of momentum in “particle physics” … but at least as far as I understand it, momentum and angular momentum are the same “thing”, angular momentum is just momentum constrained by some other physical force (such as gravity for things moving on their obits, or a gravitationally bound object such as stars or rubble pile asteroids spinning on their axis of rotation)
Perhaps I'm really being thick, but I take the answer as: if matter falls directly down onto a neutron star, that infalling mass doesn't add any spin.
If that mass comes down at an angle to the star (and it's swirling, right?), the kinetic energy gives a sideways push where it hits, like someone brushing the side of a football - it adds some rotational energy.
But that seems a trivial answer so I guess you're asking something deeper?
I may be entirely wrong, but I don't think the analogy holds, as friction is what makes the football spin.
Maybe a better one would be a spinning ice skater: going from wide arms to closed arms increases angular velocity.
Indeed consider the ingress mass and the neutron star as a single system, ingress mass gets closer (as the ice skater arms do) and angular velocity increases. Collapse is what makes it spin faster.
