As I understand it, ‘a straight wire with gain’ was a phrase that first comes from Stewart Hegeman, designer of the famous Harmon Kardon Citation II amp and several speakers, particularly omni-directional. I take the term to mean that the audio signal flows through a minimalistic circuit with the fewest possible components, thus ensuring the purest possible sound for the gain achieved. Frankly very few amplifiers do this, but it is also my own goal with my 2 stage single ended tube amps. Very simple circuits, though fed by substantial power supplies with at least 2 chokes and premium polypropylene caps.
It's a while ago, but I seem to remember Quad challenging people to tell the differences between 2 amps, implying that amps had no distinct sound of their own. A pretty wacky idea - even components sound different to trained ears. I don't know if anyone remembers this?