The question was about current and voltage. The current and voltage cannot be the same if the harmonics are temporally shifted relative to the fundamental.
Since almost agree that tubes sound different than solid state, then what makes up the ’voltage’ must vary.
If you put a solid state amp and a tube amp on the bench, both might measure quite flat- the H/K Citation 2 is a good example. One amp might sound bright while the other does not. But if you measure the output at the frequencies where the bright amp sounds that way, you'll see that the power can be the same as the amp that is not bright (likely the tube amp).
The difference in sound is higher ordered harmonics which tend to be audible with most solid state amps (audiophiles have been hearing this problem for the last 60 years, which has kept the tube industry alive), and the ear is both keenly sensitive to these harmonics (it uses them to sense sound pressure) and assigns a tonality (as it does for all forms of distortion) of 'bright and harsh'.
Yet at any given frequency, for a given amount of power into a given load (8 Ohms for example) the voltage and current will be the same on account on the power formula (1 Watt= 1 Amp times 1 Volt). This is a simple fact that cannot be undone. The difference you hear isn't on account of power! Its on account of the distortion signature, which is simply too little energy to show up when measuring voltage and current.
The big reveal here is that distortion is always audible! The ear has over a 120dB range and since it uses higher ordered harmonics to know how loud a sound is, its sensitivity to those harmonics is paramount, but poorly understood by audiophiles and engineers alike. Yet that is the main difference you hear between tube and solid state.