Take the equation $v = \omega r$. If $r = 0$, then $v = 0$. Rearranging we get $$\omega = \frac{0}{0},$$
which is indeterminate. That does not mean that $\omega$ is zero at the centre when r =0 and neither does it mean that it is something impossible like infinite. Indeterminate just means the maths cannot give us a definite answer without additional information. All we can do is is make the most reasonable assumption that $\omega$ at the centre is the same as it is everywhere else on the disk.
However, let's step back a minute and consider that we are now talking about the angular velocity of a point with zero radius. Is that reasonable? It cannot possibly be an extended particle like an atom. We are essentially asking about the angular velocity of nothing, which is a pointless question. On the other hand, we talk about the properties of an electron which is commonly said to be a dimensionless point particle, but when we look deeper, there are good arguments that an electron is not really a point particle. See this article.
Now consider this. We agree any point on the disk that is not exactly at the centre of the disk has the same non-zero angular velocity $\omega$. So where exactly, is the exact centre of the disk? Well the Heisenberg uncertainty principle tells us we can not be exactly certain of its location. If we use our best estimate or calculation of where the exact centre of the disk is and assume there is some physical but dimensionless point particle at that location that has properties like mass, then the uncertainty principle tells us the point particle is not exactly at that location, but is in a superposition of locations about that point. Statistics tells us the probability of the 'particle' being exactly at the centre is practically zero. This hypothetical, but probably none existent point particle, cannot be exactly at the centre of the disk and so it is at a non zero distance from the centre and shares the same angular velocity as the rest of the disk.