Whether power amplification, voltage amplification or current amplification matters is a question of the amplifier's application.
Different applications require different ways of specifying the amp. It matters whether you are driving a motor, an antenna, a pair of audio speakers, LED lighting, piezo transducers, etc...
I'll attempt your question from the perspective of an audio amplifier, since that seems to be your primary concern:
An amplifier can be said to have a power gain of x decibels. The gain of an amp in decibels is never voltage or current, it is always power.
An audio amplifier is sometimes called a "power amplifier" but that is a misnomer. Usually the amount of power entering the amplifier is no factor in determining the output power or voltage.
What matter with audio "end stage" amplifiers is how much power is delivered into a specified load of, say, 8 Ohm or 4 Ohm.
The input voltage level that produces that maximum output power is often also specified, so that the pre-amp or the source output signal levels can be set to deliver the desired output power.
Since I'm only given the power gain, I don't really know the output voltage of the amplifier. I can feed the amp with some known input voltage, and given the input impedance of the amp I can calculate the input power and therefore the output power.
The input impedance matters only in order to determine the input voltage once it is connected to a source or a pre-amp output. It does not matter what input power that input voltage corresponds to.
With an input impedance of, say, 10 kOhm, any pre-amp with an output of 100 Ohm to 1K Ohm will be fine as the loading by the input impedance will have little effect on the voltage level.
But aren't we usually more interested in how much output voltage we are getting from the amp?
Yes and no. We are actually interested in how much power an amp can deliver into a target load. Once we know the target load, we can determine the voltage required over that load. Once we know that required output voltage we can calculate the voltage gain based on the input voltage level that would produce that required output voltage.
I chose the word "level" to once more emphasize that with audio amplifiers the amount of power delivered at the input does not matter. What matter is the signal level, expressed in V.
Take the following example from an amplifier spec sheet:
One of the quoted models, model "800" produces 200W for an 8 Ohm speaker load.
The voltage over that load, at 200W, will be 40V.
Also according to the specs below, the input required to produce a 200W sinus is 1.4 V.
The voltage gain is 40V / 1.4V = 28.6 = 29dB (coincidentally, the factor and the dB number are close, so don't be confused)
That matches the spec above, quoting it at "29dB" voltage amplification. with audio amplifiers, that number is tightly connected to the specified load, 8 Ohm in this calculation.
Change the load, and the power will change, because the amplifier will deliver a different amount of power into a different load. Current and voltage will both change. This is the topic of load matching, a.k.a impedance matching.
(Specs and images from https://www.anthemav.com/downloads/str_integrated_amplifier_datasheet.pdf )