Now I'll take a not-very-rigorous stab at your exact question:

> what does an infinite density even mean?

In the previous reply I asked what does t=0 mean and under time-reversal marched some late time test objects towards a big-bang style singularity. Infinite density means that you can put any amount of matter down as initial values on a spacelike hypersurface at a time far from the singularity, evolve that surface in time towards the singularity, and eventually up at the singularity. The problem of the singularity is that a spacelike hypersurface there doesn't admit a sensible set of initial values, so you can't predict much about what comes out of the singularity when marching forward towards late times.

I also talked briefly about the pressure components of the stress-energy tensor.

Let's treat the stress-energy tensor as a 4x4 matrix describing the flux of i-momentum in the i-direction, and slice up spacetime into time-indexed 3-spaces. If we make a little 3d "cell" (being a little loose with terminology) then T^{00} encodes the amount of momentum sourced within the cell at t_{now} and staying within the cell at t_{immediately adjacent to now}.

Let's look at only one spatial dimension.

T^{11} encodes the amount of momentum originating to our left or right entering and leaving the cell to the left or right. Lets send rightward-going momentum into the cell from the left and mostly bounce out again to the left as leftward-going momentum, with some of the sent-in momentum sticking behind in the cell as energy (T^{00}) and some leaking out as rightward-going momentum to the next cell to the right. This is how pressure works, and shows how the stress-energy tensor can evolve as you squash things together. Momentum that is sent into a region (a cell being a sort of region) can stick around as energy. The question is, "how does it stick around"? Perhaps by increasing the frequency of the particle(s) occupying the cell.

Since T^00 usually dominates the stress-energy tensor T, and that in turn usually determines the Einstein tensor G, as more momentum-energy enters the cell without leaving, local curvature around the cell also increases. That in turn drives the split of momentum reflected back out of the cell, momentum retained within the cell, and momentum passed in another direction. This is essentially the root of the nonlinearity of the Einstein Field Equations.

The split of what energy-momentum reflects back out, flows right through, or remains within a cell depends essentially on paths of least resistance, and those depend on how the contents of the cell behave locally and also on the geometrical background. Grossly, momentum will tend to exit a cell in a downwards direction if it can, and downwards is in the direction of cells with greater energy (T^00).

At ever higher density around a cell, practically any kind of momentum entering the cell stays in the cell. As we move towards infinite density, all the "higher" (in a gravitational potential sense) cells eventually lose all of their energy (and any newly arriving momentum) in the direction of the "lowest" cell, even as we take the sizes of all cells to zero.

(For black holes rather than collapsing whole universes (or time-reversed expanding universes) only the cells inside the horizon are guaranteed to leak their entire energy and future momentum-energy towards the lowest cell.)

Unanswered questions: what quantum numbers are found in the lowest cell, especially as we shrink cell sizes? If you have a lowest infinitesimal cell holding all the sources of stress-energy, do quantum effects transfer momentum-energy from it to neighbouring higher infinitesimal cells? What quantum numbers escape the singularity? Can this happen hierarchically so as to form jets or other structures?