Regular audio signals are difficult to measure because they are constantly changing and the average voltage is low, and the meter won't respond to the peaks, but you you should be able to measure something and you should measure around 1VAC with a constant test-tone. Usually the controls are running at Line Level.
Don't run full-power test-tones into your speakers!!!!
There are digital pots but they can be tricky with audio. Some may not work with AC (unless they are biased) and you can get clicks, pops, or other noises, especially when you change volume. And you have to make sure it doesn't power-on at full-volume. You might be able to find "special" digital pots designed for audio, or there are VCA (voltage controlled amplifier) and DCA (digitally controlled amplifier) chips.
There was a chip called the LM1035/LM1036 which is a DC-voltage controlled volume & tone control. I don't think it's made anymore but it still seems to be available.
As you probably know, the regular Arduino doesn't have a DAC so it can't put-out a varying DC voltage but the analogWrite() PWM can be filtered/smoothed for an application like this.