If you picture a "traditional" signal as a single bar (from a bar graph), and picture OAM as any number of these bars, but splaying out from a center point radially into a circular form, you can see a theoretically infinite number of bars. 360 of them, spaced 1 degree apart, then 720 spaced at half-degree intervals, etc. (As I understand the article, they also spin these at certain speeds, which makes sense for practical reasons, but isn't necessary for the analogy) The problem becomes when your equipment can't distinguish between two neighbouring bars (or the transmitting equipment lets the bars leak, or something about the channel, eg turbulence, confuses matters).
So in mathworld, it may be theoretically infinite, but in the real world we'll hit limits. There will be a limit due to quantum mechanics because energy differences can only be measured to a certain precision. A limit due to physics of noise, and a limit due to what we can practically realize (and all the ugliness of dealing with a planet with a pesky atmosphere and temperature changes).
This is no different. It's pointless hyperbole.
What you're describing is a description of polarization.
As to OAM, it's a bit hard to explain because it's difficult to visualize. Think about an ordinary radio wave going back and forth like a sine wave in a single plane. This is linearly polarized light. Now, with the same frequency and phase another photon could also be polarized at a different angle, say 90 deg. to the other photon. However, you can essentially combine both aspects of waves in varying degrees, creating elliptical and circularly polarized waves. You can think of that as the photon "spinning". OAM is on a different scale. Normally we think of light as traveling in purely straight lines other than the wave effects. However, take that model of a rotating photon wave-packet and at a larger scale imagine the photon "in orbit" along a trajectory, taking a cork-screw path.
Circularly polarized light is like a spinning football being thrown. Light with non-zero OAM is like a curve ball (although admittedly the analogy breaks down).
At first glance, I though OAM sounded like circular polarization with different angular velocities. Now your comment makes me think there's more to it. Maybe someday I'll get back on track for a physics degree, and I'll be able to think through it properly.