I think there are thematic distinctions to be made with regard to how a character does something, particularly with regard to what's being used (in the context of a fight) to attack versus what's being used to assist with an attack.
That is to say, Spider-Man still ultimately punches and kicks the bad guys he faces, so I see him as a martial character. He certainly has spider-themed abilities, such as wall-crawling, web-slinging, and notable strength, but while he can use the webs as weapons/battlefield control, more often they seem to be used as methods to help him beat up the bad guy.
I run a homebrew variant of 3e. If a player came to me and said, "I want my character to basically be Spider-Man", I'd say two things.
"First, Spider Man is super powerful and, in this game, you start out as an ordinary guy with a lot of potential who is not super powerful. So your character will have to grow incrementally into being spider man starting from fairly humble beginnings. It's not just getting bit with a magical spider, and suddenly you are a demigod. Your character will have to work to get there."
And the second thing I'd tell him is, "Ok, to reliably have control over that spider theme, you probably want to be a sorcerer of some sort and you'll want to take spells like: web splat, web line, spider climb, body weaponry, combat precognition, mage armor, bull's strength, cat's grace, and of course web. And sense you are beating people up, you'll probably want to take the feat 'Arcane Strike' because it will give you very flexible combat powers. One problem I've had with players that do this sort of thing before is they eventually can't resist the temptation to branch out and do useful things that don't fit the theme they originally had. The rules don't constrain you to just picking a set of themed spells, and it's really hard to write up a set of fair rules that rewards a player for picking spells in a theme. This is not a Supers game where you can choose everything you do ahead of time and get rewarded for being a narrow one trick pony. I had a player want to play a telepath and they stuck with it for like seven levels or so and were a useful party member, but the temptation to do big splashy damaging effects got to them and they started adding in things splashier things like Shrapnel Burst. Ultimately, that's on you. I'm not going to tell you who your character is."
Likewise, I'm of the opinion that there's a difference to be drawn with regard to characters that actually cast spells, and those that have some sort of magical/supernatural powers. The former simply seems (to me) to come with a set of assumptions that the latter doesn't necessary have, at least with regard to presentation. Spellcasters carry an implication that they've studied magic for the sake of studying magic, i.e. that it's an esoteric discipline which can be academically pursued and which allows for a wide range of abilities to be acquired outside of any particular theme or grouping.
Sorcerers don't carry that baggage. Like, I think we can both readily agree that Charlie from "Firestarter" is in D&D terms some sort of Sorcerer with a pyromancy theme. She doesn't have the presentation of studying magic or academics or any of that wizardly stuff. Despite being the product of esoteric technology, she is some sort of magical/supernatural being. In Stephen King's terms, she either does or probably does "Shine".
So, if we agree on that, then let me go one further. You can imagine a wizard whose schtick is using magic to beat people up good. You don't have to use fireballs to beat people up. You could transform yourself into Hyde, use the equivalent of magical steroids to beef up and turn your skin into flexible bronze, and pound the crap out of people. While wizards maybe don't do that often by inclination or practicality, they certainly could. It's within the realm of magic.
Consider the Jedi Knights. They are certainly wizard warriors. They use divination magic to predict and counter opponent's attacks, and telekinesis magic to give them an edge over enemies. They certainly are using swords and aren't disqualified as wizards. They are not fighters/martial classes (the way Han Solo, Cad Bane or Boba Fett is).
So yeah, if you want to be Spiderman in my game, you don't go at it by being a martial class. And if you are like, "This sucks. I'm a weak Spider Man, and then I run out of magic and I'm like really vulnerable.", I'll be like, "Well, yeah, but if you were actually spider-man who had 80 times the strength of a normal human and all these powers all the time, you'd overshadow the rest of the party. Also, the spider-man you love has the power of plot. He ought to die in the first good machine gun blast if we were doing actual simulation, because he's not quick enough to dodge multiple bullets even if he gets a head start due to spider sense, and nothing about him actually implies he gained super durability."