It's a bit of both, to be honest. E.g., the Fender Stratocaster and Gibson Les Paul have very different sounds due to different pickups, electrics, tone woods, etc. People will buy a particular type of guitar because they want a particular sound. Choice of amplifier also comes into play here.
OTOH, there is definitely brand cachet/snobbery. A Gibson Les Paul will almost always get more respect than a Les Paul copy from another brand.
What I'm less aware of, except from perhaps a few niche guitar makers I know, is brand snobbery of the form of, "Well, Fender are better than Gibson and Gretsch because blah", or vice versa (people do tend to be a bit snobby about Ibanez though, for some reason).
Most people I know buy premium guitars (and that's a loaded term in itself) for the sound rather than the brand... except that, as I've already suggested, they will often prefer a well-known brand over a "knock-off" or lower end brand: so "serious" musicians will generally buy Fender rather than Squire (low end brand owned by Fender).
Different genres of music can lead to different brand preferences. More "classic" music genres tend to favour Gibson, Fender, Gretsch, Rickenbacker, and so on, whereas newer or more experimental genres tend to favour Ibanez, ESP, PRS, Jackson, etc. This is more of an observation than a hard and fast rule though, and there will always be exceptions, like the Fender Jaguar which has been a firm favourite of some more alternative bands since the late 80s.
Hopefully that rather rambling answer makes some sense.
There are distinct sound differences, although they may be less obvious to a layman, the same way functional programming vs OOP would just "look like code" to someone who was not a developer.
For example, Fenders, especially models like the Telecaster, have a "twangier", more treble-forward tone (they're used a lot in country music), where as a Gibson model, say the SG, would have a "meatier" more bass-forward tone and generally longer sustain (used more in rock, e.g., AC/DC is a good example).
Another big difference is the scale length between a Fender and Gibson (the length of playable string between the nut and where the string contacts the bridge saddle).
On a Fender Telecaster, Stratocaster, or Jazzmaster, it's longer than on, say, a Gibson Les Paul or ES-335. This means there's more string tension and "snap" on those Fenders, which some players prefer. J Mascis of Dinosaur Jr (who plays a Fender Jazzmaster) says, "I like to be at war with the guitar." It also makes the guitar a little harder to play and a bit less forgiving when you make a mistake.
The original models of the 50s and 60s sounded very differently. Modern incarnations of the old classics (Telecaster, Stratocaster, Les Paul, SG) are still produced and sold in large numbers, but it's mostly for the brand, as you mentioned.
Nowadays modern electric guitars are much more versatile. They generally can emulate the sound of the classics easily, but they play differently, they lack their slow, thick necks, their poor tuning, their noisy pickups... :)
They absolutely sound different. A Fender Jazzmaster doesn't sound like a Fender Telecaster, and even within the same model, a Squier won't sound like a guitar build in US. A good guitarist can tell. But the most important things is that they also play differently. Some models are better suited for some genres too. This is mostly true for every acoustic or electric instrument by the way. The brand actually matters.
BUT a lot of guitarists also forget is sound is as good as the weakest link in the sound chain.
A $10.000 electric guitar will not sound good on a shitty amp.
Guitars that are substantially different in terms of construction or components do sound different from each other. However, electric guitarists have access to a tremendous variety of tools to alter their sound, so the differences in the basic sound of the guitar might be dominated by effects pedals or choice of amp and gain/EQ settings, or just simple things like which pickup you're using or how close to the bridge you pluck the strings.
