If you tried to extend the notion of position to the singularity, it would become discontinuous. You could extend time. The singularity could be given a time coordinate. But that does not give you a coordinate system (a "manifold") that extends to include the singularity.

If you have two objects traveling around a circle and they collide, you can say they collided at a point and a chosen coordinate system can identify the location of the collision. If instead the circle shrinks to zero diameter, the coordinate system itself disappears when they meet.

The Wikipedia article on the concept of a singularity uses traveling North on Earth as a similar example: when you get to the North Pole, the concept of longitude becomes meaningless.