I don't have much to offer in terms of experience with headphones but in terms of my home listening systems there's a large difference between the various receivers I have which are all SS and are of various age ranging from 2 to 40 years.
As an electrical engineer I can tell you that yes, they are all functionally equivalent but there are lots of other factors including the transistors themselves. Topology is an obvious difference but lets not even consider that.
Besides, if they all sounded the same then why on earth are there so many companies making amplifiers..... Also, if you follow your own logic all DACs are the same too. They're functionally equivalent given the same speed/bit spec.
I don't agree that they all sound the same because as we're finding out, plenty of the people making them don't know what they're doing and are doing stuff that actually harms the sound. Competently designed amps I think would likely sound very similar, very possibly would sound the same within the range of human hearing, but I don't know how many products that's even true of. Very few of the boutique products it seems.
Because people are not actually measuring them. Not to mention comparing them in different circumstances or based on memory or any number of other things that skew their findings.
You make a mistake by saying that your amps are functionally the same, which they almost certainly aren't, unless you've somehow managed to get a variety of them that have the same measured output and are at similar levels of duress (i.e. clean, which is of more significance for older ones). It actually doesn't need to be the same, they just need to all effectively have the same headroom such that whatever volume levels you're comparing them at, they are not running into issues with being able to provide the necessary output.
The reality is that not all amps are competently designed (and by that I mean measure such that they have no glaring fault that would be audible), and plenty of these companies often even tout "voicing" them or "engineering them by ear". Or at least a lot of the headphone amp makers do. Stuff like output impedance issues are actually ridiculously common in headphone amps, and even intended plenty of the time, and that alone can greatly impact the results you get.
Even if they do sound the same, people want/expect them to sound different and so will convince themselves that they do.
Well, that's actually true. In fact, I'd say DACs are more likely to sound the same than amps. Of course there are things that can skew comparisons between DACs, so you'd want to take extra care to remove all other factors (so you'd want same amp/speakers/measured volume/recording/etc).