A lot of great stuff has already been said, but I'll chime in anyway.
Amps sound different from each other for all the reasons already mentioned - parts choices, "architecture" or arrangement of gain stages, order of building blocks, e.g. is the tone stack after the first gain stage (Fender BF) or after the entire preamp (Marshall), phase inverter style, tube types, etc. ad nauseam. Did anyone mention iron? Transformer construction can make a big difference.
Anyway... I'd like to add a couple of brief points, then get into the nitty-gritty of tone shaping, which I think is more to the point of the original question.
First, a lot of amps
do sound very much the same, particularly amps based on other amps. In fact, that's the point, isn't it? Play any five 100w plexi clones thru the same cab, and they will sound nearly identical, with minor variations due to parts choices. Same for JCM800-based amps, BF clones, tweeds, Dumbles, Trainwrecks, etc.
Second, not all all-tube amps use 12AX7s. Really? Yes!
The issue of preamp vs. power amp is a huge one. OP, you are right in that most amps do their tone-shaping in the preamp, and after the Master Volume, the power section just makes it bigger. But that is so sad! There is so much tone and flavor and feel and crunch and sex to be found in overdriving the power section, you're really missing out if you don't do it. There's a ton of debate on that, so I'll just leave it there.
Now, let's get down into the nitty-gritty of tone shaping within an amp. Putting aside the "architecture" of the amp, there are a few main places where amp builders make component choices that greatly affect the tone of the amp. There are others, but these are the main ones. The cathode R/C (resistor and capacitor) of each gain stage and the coupling between stages.
On the cathode of each gain stage, the lower the resistance (to a point), the higher the gain of the stage. Adding a capacitor adds gain above a roll-off point, the frequency of which depends on the value of the cap and resistor.
Between stages, you have a coupling cap. A smaller cap cuts more bass. Then you might have a voltage divider comprised of two resistors, the value of which determine how much signal hits the next stage. Part of the voltage divider might be bypassed with a cap, passing more highs thru to the next stage. Finally, the grid resistor of the next stage (which might or might not be part of the voltage divider) has an effect on tone. Very high values bleed off highs.
How all those things are handled at each stage will affect the signal down the line. For example, if you thin out the tone at the first gain stage, you can push gain much higher, then you can get some of the "thump" back later (if you do it right - it's not as simple as it sounds). If you want a big clean sound, you need to keep the low end intact. Want chime? Boost the very high highs. And on and on.
Like I said there are other places in the circuit that allow for tone shaping. A lot of high-gain amps have small capacitors across the plate resistors to bleed off highs (or is it to fight parasitic oscillation due to poor lead dress or PCB design, wink, wink?).
Lastly, I'd like to make the case for simplicity. The more circuitry your guitar's signal must pass thru, the less pure it becomes. If you put all kinds of tone-shaping crap around every gain stage, especially if you have a whole bunch of gain stages run together, you're going to spoil the purity and clarity of the sound. That's why so many amps have no "soul." They might have bells and whistles up the wazoo, but something is missing - you know what I'm talking about. Sometimes the best tone shaping at a particular gain stage is none. Pick components that give the amp the voice you want, keep it simple, and you'll be surprised how good it sounds.
I hope this helps.
