Does gear define the player?
I don't doubt that some players have "stumbled" onto some combination of gear which inspires and defines a signature sound. But I think you have to look at the player's predisposition. Was finding that sound an accident, or did he have the good luck to figure out how to make the kind of sound that he wanted to hear?
There's a tremendous amount of complexity hidden just below the surface of a good musical instrument. I think it's a huge mistake to assume that a certain kind of instrument is tied to a particular style of music or sound. The way the player approaches an instrument has a huge impact upon what comes out of it. Consider the guitar: small changes in picking-hand position and technique can cause significant changes in timbre. Likewise, small changes in picking intensity will have an effect upon the sound that comes out of your amp.
A comment attributed to Pete Townshend about the need to match your guitar to your amp, etc. is a good starting point, but it doesn't go far enough. I think a good player will examine all aspects of his rig and try to optimize it for the way he wants it to sound and respond. It's the combination of all the decisions regarding gear and all the nuances of technique that define a player's sound. You might get 80% of the way with just the player or just the gear, but it's the combination that really defines what the listener comes to associate with the player.
There may be, if you look at groups of players rather than individuals, certain tendencies depending upon age or social group. Obviously there are going to be rather broad generalizations. But I have to wonder at what age it is that the musical "imprinting" really takes place.
I grew up in a family that was musical on my mother's side. Most of the adults on mom's side of the family were classical musicians who performed and taught. Some of them played multiple instruments, but they all played the piano. So I grew up around a lot of classical piano music when I was very young. In my preteen and teenage years (early `60s through early `70s) I listened to a lot of what was "in the air", but gravitated toward music that had (for its time, of course) a more "modern" sound.
Now that I'm about eight years into having renewed my interest in playing guitar, I find that I can play a lot of the kinds of music that I've always imagined, and it comes out with certain classical elements regarding voicings and composition. But the sounds are very much from the era of my youth. So it's really an amalgamation of two very distinct influences - the music being quite separate from the sound. That's not to say that I won't throw in a gratiutous Jimmy Page-ism or that I don't enjoy a nice piano-like clean tone from my guitar... but they're not the primary elements of my playing or my sound.
On the other hand, my playing is colored by what I'd call "jazz-isms". I don't think of myself as a jazz player and don't want to be a jazz player, but there are some aspects of a jazz style - in particular the "swing" of jazz - in some of my playing. Oddly, I never listened to jazz in my youth and still don't have a deep appreciation for the genre. But I'm learning now that many of my favorite players came to rock from a background which included jazz.
Although the gear that I use now is physically quite different from that which I used and admired during my musically formative years, the sound is largely of that era: a clean to slightly-overdriven base tone with some reverb, sometimes echo and occasionally a modulation effect. It's not so much that I won't use any other tone - I will always try to find something that fits the song - but when I'm playing for myself I always start with the tones that were "imprinted" upon me when I was younger.
So from my perspective (for whatever that's worth) the playing and the tone are pretty much inseparable. My main rig today bears a strong resemblance to the first rig that I really was comfortable with all those decades ago. But I think the reason that I've gravitated toward a similar rig is because my musical "vision" has, at its core at least, remained fairly constant (although, thankfully, my ability to realize that vision has improved immensely). I think that had some other musical style grabbed my imagination in the meantime, I would have searched for a comfortable rig to support that style.
All of which is, ultimately, a very long-winded way of saying that I believe it's the player that defines the gear, and not vice versa...