Mach's principle is a very deep question that has evoked controversy between the likes of Newton, Leibniz, Clarke, Einstein, and so on, and so far as I understand it not entirely resolved to everyone's satisfaction. That said, there is a simple perspective on the question that might help.
Most of the people invoking Mach's principle did so in a pre-relativistic era and the relations (angles and distances) between objects they were considering were purely spatial. In other words, they were assuming a Galilean geometry. You can only measure angles or distances between objects taken at the same instant of time. If you are on a moving train, and throw a ball in the air and catch it, the distance between where the ball starts and ends is zero for the observer on the train, and something much larger for the observer on the platform. Distances are only fixed in each timeslice, but the timeslices can slide around relative to one another as we change reference frame, so distances between timeslices are undefined.
This is not the case in relativity, which is much more rigid. In Minkowski space, we can measure well-defined spacetime intervals between events that (for most observers) occur at different times. The timeslices that in Galilean relativity are free to slide around relative to one another are cross-linked by rigid distances in Minkowski space.
If we want to apply Mach's principle and make physics depend only on the distances between objects, in relativity we have to do this between events at different times, because for some other moving observer they will be the same time. This gives us much less freedom.
Now, it so happens that translations and rotations (including boosts) of spacetime preserve Minkowski distances. However, switching from a stationary reference frame to a rotating frame has the effect of applying a helical twist to spacetime, and this does not preserve Minkowski distances between spacetime points in different timeslices.
If we consider a rotating disc, the circumference shrinks along the direction of motion by relativistic length contraction, but the radius of the disc remains the same. The planes of simultaneity of points on the circumference are tangent to a helix (like a corkscrew) in spacetime. The 'now' on the circumference does not join up with itself when you move all the way around the circle. So a rotating disc does not and can not have the same spatial relations between its points as a stationary one. They are fundamentally different, geometrically. You can take the stationary one and describe it in a rotating coordinate system, so that it looks point-to-point identical to the rotating disc, but this system necessarily has an intrinsic curvature, as it has to distort some of the across-time distances. You can consider this curvature to introduce 'fictitious' forces - centrifugal/Coriolis forces for rotation - like curved spacetime around a mass creates gravity.
This is related to the idea of Born rigidity and the Ehrenfest paradox - rigid bodies don't work well in special relativity. You can, with some difficulty, accelerate an object linearly and maintain all inter-particle distances, but you cannot generally rotate an object rigidly, without distorting it.
So, the direct answer to your question is that you measure the angular velocity of the disc relative to itself in its own past and future, as well as the present.
That's far from being the end of the story. In general relativity, we can ask about why spacetime distances work this way. And we have phenomena like frame dragging in which the motion of gravitating objects twists spacetime to rotate with it, that opens up all sorts of new questions. It is, as I said at the start, a very deep subject.
