Does matter accelerate to the speed of light as it approaches the singularity? $$v_\mathrm e=\sqrt{\frac{2GM}r}$$

If we understand the escape velocity to be the speed needed to escape from the 'surface' of a gravitational object described by the above equation.

It is equally true that the escape velocity is the speed an object pulled from rest will reach when it hits a certain point in the gravitational field.

Since the event horizon is defined as when the escape velocity reaches the speed of light for a black hole, would matter accelerate near the speed of light as it approaches the singularity? Accretion disks are known to move at relativistic speeds near the event horizon, but perhaps oddities occur beyond this point?

The event horizon can be derived from setting the escape velocity of an object to the speed of light, it is a point of no return. I understand there are better definitions, but this is not really germane to the question at hand. How about from the perspective of another in-falling observer?
There is indeed a sense that infalling objects accelerate to the speed of light as it reaches the event horizon, but you need to be careful what you mean by the speed of the object because the speed is observer dependent.

This is explained in detail on the Physics site in my answer to Will an object always fall at an infinite speed in a black hole? An observer watching from far from the black hole would see the falling object initially accelerate towards the black hole but then decelerate to a halt at the event horizon. However an observer hovering a distance $d$ above the event horizon would see the falling object pass them at a speed:

$$ v = c \sqrt{\frac{r_s}{r_s + d}} \tag{1} $$

where $r_s$ is the radius of the event horizon. As the distance above the horizon $d$ goes to zero the speed calculated from equation (1) goes to the speed of light $c$.