"Center of a black hole is a time"

The Schwarzschild metric as you've written it is only one particular coordinate system and the fact that $r$ and $t$ switch roles at the event horizon is an artifact of that coordinate system. There are other coordinate systems which make certain properties of the metric more intuitive. The ones that might be most useful for you are ones which can be drawn as a Penrose diagram. In a Penrose diagram time is always up and radius is always side to side and light always travels at 45 degrees. On the diagram of a black hole you can see that the event horizon is just the point where all light (and therefore everything) must hit the singularity because the singularity encompasses all of its possible future positions.

What the author means is as follows. Consider the (un-normalized) vector field $\partial_r$ where $r$ is the radial coordinate; $\partial_r$ is thus just the vector field orthogonal to the level sets $r = \text{const.}$ or, equivalently, it is the vector field foliating said level sets.

As an aside, note that Schwarzschild coordinates are perfectly valid strictly inside the horizon as well as strictly outside, just not on the horizon itself; thus we cannot use them if we want to describe processes that evolve from the exterior to the interior.

Now $g(\partial_r,\partial_r)$ is a coordinate-independent quantity so it doesn't matter what coordinates we use to calculate it in the interior. We find in said interior that $g(\partial_r,\partial_r) < 0$ meaning the vector field is time-like. Thus the surface $r = 0$, which is a level set of the vector field, is a space-like surface (a space-like surface is by definition orthogonal to a time-like vector field). What this means is the singularity $r = 0$ is a moment in time as opposed to a point in space.

This is analogous to looking at a $t = \text{const.}$ surface in, say, a global inertial frame in flat space-time and interpreting it as the simultaneity surface of the family of inertial observers at rest in the frame, with clocks that are all synchronized. However it should be noted that in the Schwarzschild black hole interior, the family of observers corresponding to the $\partial_r$ vector field can't synchronize their clocks if said clocks are set to read proper time because of the relative motion of these observers; they would have to adjust their clocks accordingly.