What is the name of this vector quantity that is similar to angular momentum? Consider 2 identical circular disks, rotating with angular velocity $\omega$ and $-\omega$ respectively along the same axis. If we declare the mass of each disk to be $m$ and radius to be $r$ then the total angular momentum of the system is of course:
$$ mr\omega - mr \omega = 0$$
Nevertheless, we can tune the pair $(mr\omega, -mr\omega)$ to ever larger numbers (or even go the opposite direction and the disks rotate opposite to how we started) and so this pair of numbers behaves like a one dimensional vector, but it can't be classical angular momentum. So what exactly is it?

A More physical realization: 
Consider the image here:

If we make an identical copy to it, and place it so the big gear is above facing down on other big gear, and little gears opposite each other, moreover if we change the radius and mass of the gears so they are all the same, then we again we can have a system with 0 net angular momentum.
 A: Two unconstrained disks have six total angular momentum degrees of freedom (because the angular momentum of each disk can take on any value and point in any direction, independent of the other). The situation you have described is one in which the two disks are subject to constraints. 
First, their angular momentum vectors are constrained to point in a particular direction (let's say that the second disk is constrained to lie along whatever direction the first disk happens to be pointing), so that removes two degrees of freedom from the system, leaving us with four (the magnitude of the first and second disk's angular momentum each count for one, and the direction that the disks' axis of rotation is pointing accounts for the other two).
In addition, the magnitudes of their angular momenta are constrained to be equal and opposite (there is nothing special about them adding to zero; this logic holds equally well if you constrained them to add to anything else). So that removes another degree of freedom from the system, leaving us with three (the magnitude of the angular momentum of the first disk, and the direction that the axis of rotation is pointing). Since the moments of inertia are assumed to be constant, we can also say that the angular velocity magnitude $\omega$ is a proxy for the angular momentum magnitude.
This leaves us with a constrained system that has three parameters: the angular velocity $\omega$ and the angles $\theta$ and $\phi$ that define the orientation of the axis of rotation (well, there are actually two choices of $(\theta,\phi)$ that will define the same axis, so there's a degeneracy here that needs to be addressed with a right-hand-rule-style choice). The orientation of the axis of rotation doesn't appear to be relevant for this situation, and ignoring those parameters is equivalent to implicitly fixing the axis to point in a certain constant direction, so we're left with one free parameter.
Overall, what you've done is as follows: you have imposed constraints on the system such that it has only one degree of freedom, $\omega$. Your choice of constraints is also far from unique, and the choice of the disks to lie along the same axis and spin with equal and opposite velocities is not particularly meaningful, in this sense. A system in which I constrained the disks' axes of rotation to be exactly 57 degrees off from each other, and in which I constrained the sum of the disks' angular speeds to be precisely 123 rad/s, would give you exactly the same behavior, in that this system would be parametrized by a single unconstrained degree of freedom $\omega$. (The constraint forces that keep this system in line would be different, but all they do is enforce the constraints I imposed on the coordinates anyway, so we can focus on how the coordinates behave under the constraints. This is one of the reasons why Lagrangian/Hamiltonian mechanics is so nice: you don't have to go mucking about actually figuring out what the constraint forces are, as long as you know what constraints they're supposed to apply.)
The profound part of this is that the system you made and the system I made obey the same dynamics. Even if I were to choose a different definition for $\omega$ (say, instead of choosing the magnitude of the angular velocity of the first disk, I chose the magnitude of the translational speed at the edge of the second disk), the systems could still be transformed into one another with a suitable change of coordinates, because fundamentally, they're just two representatives of a much larger class of systems: systems with one unconstrained degree of freedom. Both your system and mine are mathematically identical to a free particle moving in one dimension.
