Most touring players use a variety of amplifiers, depending on where they're playing because it's not always practical to take heavy amplifiers to other countries. I know someone who has played with Guthrie Govan and Robben Ford in several places and he didn't necessarily use the same amps, but he's another one who sounds like himself and IMO, it's because he/they know how to get 'their' sound from a variety of equipment. Ford doesn't always use a Dumble- I have seen videos of him using Twins and other models.
As far as tonewood, I'm in the camp of believing that it makes a difference. Listen to an electric guitar played without amplification and the same models will sound similar, but different. Played with amplification, they still sound different, even if the same pickups are used because the strings don't vibrate the same when different woods are used and there are too many variables to definitively say that they're different or the same. However, by eliminating the variable of how the string is picked or plucked, the waveforms can be compared.
Even in acoustic guitars, the various lumber species sound different but then, trees aren't consistent. Lumber species have similarities, so they're used for specific purposes but I'm seeing a lot more variety than I did when I worked at a music store- at that time, it was acoustic guitars with Brazilian/Indian Rosewood, Mahogany, Maple bodies with Sitka or Adirondac Spruce, Mahogany, Cedar and a few other species for tops. Electrics were more varied. Each has its own sound.