Electronic components do change with temperature and also capacitors 'form' with use. In fact if an electronic device isn't used for a period of time, the capacitors will lose their internal 'shape' and become out of spec; they need to be reformed by bringing the device up on a variac from low voltage over a time frame of days sometimes to avoid damage.
So, with that being said, I can see where they speak of 'breaking in" an amplifier. I think it's probably more a factor of burning in the components inside.
I definitely hear differences between the amplifiers; they each have a specific sound which of course would be defined by what the design is internally and the quality of the components used. Some use very high quality high tolerance devices, others do not . I guess price dictates what you get to large degree.
