The effect is
design dependent. So the topology and circuit around them determines the 'sound' of the amplifier as a whole.
One can design a tube amplifier with just 1 tube, say a simple triode that has to do all of it. That tube is very flawed in specific ways and alters the signal. One can build an amplifier that has properties that change depending on the inserted tube.
Frequency response, distortion, gain, noise will all be different.
The fact that the gain can differ (even between channels) is often the culprit of someone perceiving the tubes to be different sounding.
Also frequency response and noise levels can be a reason to perceive them to have a different sound.
The problem is to determine if the change in sound you hear is real (and not caused by say the gain or expectation) is that to determine that there really is an audible difference you would have to test this in a scientific way.
This means: having 2 identical amplifiers, The ability to adjust the gain/volume of each channel, using 2 different tubes in each amp, let both amps warm up say. 5mins, and then switch (blind) between the 2 amps. Only then you can truly determine IF the tubes sound different or that it is 'something else' causing the perceived differences.
I am quite certain you have never done that. You can be quite certain I have with a bunch of tubes in 2 equal amps in such a setup because I have a few of those amps (proto's, pre-prod and 1 prod unit) and I can tell you that my non audiophile ears (no idea what Harman level I am
) can't hear differences with quite a few tubes that are well suited but there are also a bunch of tubes that roll-off in the audible band or have a lot of hiss in that circuit and can easily distinguish them.
I don't see that as a having a sound but rather these tubes being faulty or not really usable in that particular design. Of course there are owners who love those tubes (rolled-off or a low added noise level)
You call it 'tubes having a sound' and I call it tubes altering the signal in specific ways determined by the used topology, schematic and tube.
One can also design an amplifier using a bunch of tubes but uses overall feedback. In that case the sound you hear with specific tubes in the simple design will not be there any more. Same tubes being swapped no change in sound. So the same tubes that have a 'sound' in circuit A may not have 'a sound' in circuit B.
It's not the tubes that have 'a sound' it is the circuit and topology combined with specific tubes that can change the response in specific ways (and thus the sound). The fact that you may prefer certain alterations in sound in a certain way is another matter.