The question of whether Lee and Ditko intended Spider-Man to be Superman is secondary to the OP's argument that Spider-Man is similar to Superman. It's only brought up as an argument against it, but that's neither here nor there. The thing about Superman and Spider-Man is also the thing with Marvel and DC, in that it went two ways and both directions. Marvel in the '60s took inspiration from DC but sought and went in a more contemporary direction. But as Marvel became big, DC took inspiration from it.
If we ignore the whole "AF#15 is the first and only important Spider-Man story" ever argument, if you read ASM's first 10 issues, where the Daily Bugle becomes the center of Peter's world (Peter being in high school and a teenager was far less important in that time, than him being poor and needing to work a crappy job for Jameson). And the Daily Bugle and Peter's entire dynamic with Jameson is quite obviously inspired by Superman and Clark Kent. And some of the high satire and parody seems to actually borrow itself from Harvey Kurtzman's magnificent satire Superduperman which also shows Clark's connection with the media and superheroics in a pretty sketical and ironic light, and EC Comics was a huge inspiration on Stan Lee. So Spider-Man took inspiration from Superman, and also Batman when you consider stuff like Peter's use of gadgets and inventions, him putting stuff in his belt, and wondering if his belt can accommodate pockets for extra cartridges and so on.
But Spider-Man and his stories gradually inspired both Batman and Superman. Like for instance if you read Strange Apparitions by Englehart where you have Bruce Wayne in a more vulnerable place worried about how his work as Batman is getting in his way with his relationship with Silver St. Cloud, and it's hard not to see Bruce act a fair bit like Peter Parker. Of course that trait was phased out when Frank Miller arrived. Everyone says that the Green Goblin is a Joker knock-off but people forget that Goblin killed Gwen Stacy ten years or so before The Killing Joke sent Joker on open season on Batman's supporting cast. And in a lot of ways, Green Goblin did the stuff that Joker later originated. DC as a rule tend to adapt and borrow (and steal). So Lex Luthor was once a red-head generic bad-guy until Captain Marvel did well and they made him like Dr. Sivana, and then later people liked Fisk, so they made Luthor into Kingpin. For a long time, the idea that Lois knowing and supporting and accepting Clark and Superman was anathema and so on. And Spider-Man and MJ proved them wrong.
And yes, I know people will cite Flash and Iris, while I can go back and cite the Wesley Dodd Sandman and Dian Belmont. Flash wasn't all that big after about the mid-50s and Iris West was dead by the time of the 80s, and Barry Allen was considered expendable by DC for COIE. So he wasn't a major hero by then unlike Spider-Man was. Heck Reed and Sue and Hank and Janet aren't unique if you factor in Hro Talak and Shayera Hol as Hawkman and Hawkgirl. Or you know Jane Thor...a civilian love-interest becoming a superheroine, quel horreur...except you know Shayera Hol was introduced as a civilian love interest to Hawkman before being made by Gardner Fox into her partner, and now someone who is bigger than Hawkman.
You know the biggest impact Spider-Man made on Superman is evident when you look at Bruce Timm's
Superman the Animated Series, where he basically makes Lana Lang into Mary Jane Watson. A party girl who cottoned on to Clark Kent's identity back in college, knows his identity and keeps his secret and helps him out. And take a look at her design.
Bruce Timm lana lang.jpg