Why are porcelain teeth strong?

There are several things you need to consider.

First - the different between toughness and hardness. When a material is hard, it can sustain a very large stress before elastic deformation takes place. This is what is measured with the Vickers hardness test - which is one of the ways that Mohr's hardness is determined. All porcelain is hard, although different grades of hardness do exist.

The second property is toughness. This relates to fracture initiation and propagation. Fracture typically starts at a point of stress concentration - where the shape of an object makes the local stress greater than the average stress. This is clearly demonstrated with plastic bags containing potato chips, for example: there is often a small "nick" in the plastic, and if you tear the plastic at that point it's easy to rip. This is because the tip of the nick is a point of stress concentration. Try to rip the bag open elsewhere, and it's much harder.

Tough materials typically have two properties: there are not many points where stress concentration can occur, and more importantly, when a crack starts to grow there is something about the structure of the material that stops it from propagating. For example, fibers in a composite material not only add rigidity (stiffness) but the also prevent the propagation of cracks.

Finally, in the example of dropping a porcelain vase, the stress of the impact is greatly amplified by the shape of the vase (impact is some distance from the center of the vase, resulting in a large shear wave; but this wave has to propagate through a thin shell of the vase).

In general, making a ceramic tougher is done by using very fine powders, and using techniques that minimize the presence of bubbles or other stress concentrators. For porcelain teeth, the volume is quite small so one can afford to work carefully to minimize the presence of imperfections. This is harder (and more expensive) to do on a large scale.

An interesting detail pointed out by @starrise in a comment - I'm adding to the answer :

To add to this answer: zirconia (ZrO2) is metastable in the tetragonal phase at STP, and will stay in that phase for very long times as part of a polycrystalline ceramic composite (5% degradation over 10 yr in mouths). Cracks near a t-ZrO2 grain will induce a volume-increasing phase change to a monoclinic structure, inducing compressive stress around the crack tip, increasing energy required to advance the crack. The overall process toughens a polycrystalline ceramic. This method is used to toughen dental porcelain.
Just in case, and to clarify, it's only metastable "baked in" as part of a polycrystalline structure, whose mechanical constraints keep it in tetragonal form, or when in solid solution with a stabilizing element such as yttrium or cerium. And you're quite welcome! Look up tetragonal zirconia polycrystal (TZP) and/or zirconia toughened alumina (ZTA) for more information

All these things matter. It's not a complete description but I hope it helps a little bit.

A few links:

interesting description of the structure of the tooh and material properties

Letter to Nature on using tetragonal zirconia to achieve tougher ceramics