What exactly is the time constant of an ion channel gating?
Although the membrane time constant is very important in neuroscience, the time constant described in the original question is about ion channel gating rather than the rate of change of membrane voltage.

The "tau" when referring to ion channel gating is indeed referring to how long it takes to open/close.

When the membrane voltage changes, there is a new equilibrium value for the probability of the channel to be open versus closed (note: there can be other states for channels too, but for this explanation I will stick to open/closed). The rate at which this equilibrium is approached from the previous open probability is described by an exponential function, with "tau" as the time constant of that approach. A small value of tau means that the new equilibrium is reached very quickly. This site has a good explanation of the terminology and gating process, as do many others.

Time constants are involved in any system described by an exponential (these are called "linear time invariant" processes), whether decay or increase, and it is equal to the time it takes to progress approximately 63.2% of the way to the equilibrium. The cool thing about time constants (and linear time invariant processes in general) is that it doesn't matter what your starting point is or how far into the process you are, the time constant stays constant and every time "tau" passes you are now 63.2% closer to equilibrium than you were at time t-tau.