Can a variac be used to maintain constant voltage?

There are also Constant Voltage Transformers based on transformer saturation effect http://www.lenco-elect.com/constant_voltage.htm

The problem with them is poor efficiency and often some distortions of the sinewave.

Even if your voltage changes by 10%, your amp has, most likely some form of feedback making it regulated. Only max peaks will be affected and only if voltage drops. Lets assume drop of 10% making drop in available (peak) power of 20% it will change perceived loudness only by 5%. Just set your loudness not higher than 95% (to prevent clipping) and regulation will take care of the rest. Preamps, CDPs etc are all regulated.
Perceived loudness = k^(1/3.5) where k is ratio of powers.

Regulator will make it worse, in my opinion, since amplifier's supply current is in form of narrow spikes of high amplitude causing voltage drops - unless you use greatly oversized transformer or device that provides line AND load regulation.

So, if an ordinary variac does not maintain voltage at its fixed setting, what is the purpose of using such a variac, i.e. is it simply to revive an old amp from storage for instance by slowly increasing the voltage to nominal levels?

That would be good application. They also bring supply voltage to regulation range of electronics in case of wide variations (constantly too low or too high).

Variacs are not specifically designed for audio applications, but rather for industrial, and general electronic testing. They operate very much like a large potentiometer, only- using inductance, rather than resistance, to divide voltage. Further- like a pot; they can only reduce voltages and cannot increase what is presented at their input, or maintain a set voltage, if a drop occurs at the line.

Actually, a Variac is a variable Autotransformer, which can be, and often is, designed to step voltage up as well as down. That would be done as shown in the first figure of the linked writeup, where the input voltage is placed across a segment of the winding that encompasses fewer than the full number of turns. Variacs that are designed for 120 volt single-phase inputs are commonly designed to output up to 140 volts, and sometimes more.

One other benefit they might provide, in addition to those Kijanki mentioned, is some reduction of high frequency noise that may be present on the incoming AC. However, as Rodman indicated Variacs are not specifically designed for audio applications, and an audio-oriented power conditioner (which in most cases won't regulate voltage) or a power regenerator (which will regulate voltage) can be expected to be a better alternative, albeit at a considerably higher price.

Regards,
-- Al