By "warmer" sound, it is meant that, even if the tube amplifiers normally produce greater distortions than many of the more modern amplifiers, for some reason this kind of distortions sounds more pleasant than other kinds of distortions.
If some amplifiers would be ideal, producing no distortions, they could not be distinguished in any way, regardless what devices have been used to implement them. The modern Class-D amplifiers can behave almost ideally when amplifying low-frequency signals, but for the signals with frequencies close to the upper limit of the audio range their behavior becomes much less ideal.
While I have pleasant memories about a tube amplifier that I have used many years ago, I have never made comparative tests with other amplifiers, so I could not say whether it was better or worse regarding "listening fatigue".
Even when comparative tests are done in an identical setup, by swapping a tube amplifier with another amplifier, it is extremely difficult to disentangle which audio differences are caused by using tubes as the amplifying devices, which are caused by the audio transformers included in most tube amplifiers and which are caused by the interaction between the amplifier outputs and the loudspeakers, which can be very different between different kinds of amplifiers, due to their different relationship between output impedance and signal frequency.