What material are old coaxial cables made from?
It's the same stuff as other electrical cable insulation. And just as that, the structure of the stuff changes, plasticizers evaporate or decay, and the thing gets brittle. So, probably, vinyl.

Good news is that if it's stiff, the plasticizer is largely gone, so now you have stiff, brittle PVC that doesn't off-gas anymore. Also, you'd want to reach the EU reports that were written before the REACH directive, which now severely limits the use of many plasticizers that were popular back-in-the-day. It's not likely that you or anyone in your household is under acute risk of anything due to having lived in a house with a coax cable.

Downside of not containing any plasticizers anymore is that it's brittle and won't fulfill its role as isolator anymore. Also, depending on the construction and use of the cable, it might have had a barrier function against water vapor, which tends to damage the dielectric (the white stuff between inner and outer conductor, not the stuff outside the outer conductor of your coax).

It probably strikes you more on coax cables than on the usual power cords because it's a rather thin layer directly on a thick-diameter metal core, as opposed to the 2 to 5 mm of total polymer until you strike the copper of a power cord.