Obviously, touching both pins of the capacitor doesn't charge it so that's not going to cause any problems. For this to be a problem you would need to have one leg of the capacitor connected to ground and the other isolated until touched by your finger. Otherwise there is no circuit for the current to flow through.
In both JS-001-2012 and MIL-STD-883H the charged human body is modeled by a 100 pF capacitor and a 1500 ohm discharging resistance. During testing, the capacitor is fully charged to several kilovolts (2 kV, 4 kV, 6 kV and 8 kV are typical standard levels) and then discharged through the resistor connected in series to the device under test. Source: Wikipedia, Human body model.
Taking worst case from the model and, say, a 100 nF capacitor under test, we have the following situation:
simulate this circuit – Schematic created using CircuitLab
Figure 1. Equivalent circuit.
Running the simulator shows that on transfer of SW2 C1 will discharge into C2 until the voltage on each is 8 V.
I haven't the energy - and possibly the wit - to prove it mathematically.
If you connect one terminal to ground and touch the other terminal with your finger, why doesn't it damage the capacitor?
It could if the capacitor is small enough but it doesn't generally seem to be a problem in the industry.
Given static electricity normally has small amounts of current, if you repeatedly statically charge the capacitor, will it eventually fail?
Probably, but it would be difficult enough to contrive that situation.
If so, shouldn't a capacitor also also have a current rating as well?
Yes, indirectly. The internal ESR (equivalent series resistance) will cause some heating. This can be a particular problem in power supplies where significant current is flowing in and out of the capacitor on each rectifier on/off pulse. The ESR generally increases with age until the resultant voltage drop is enough to cause circuit malfunction.