The comics' explanation is that Peter isn't just a clever tinkerer, he's a genius. He has a gift for scientific invention.
Then, like Reed Richards and Tony Stark, they should show him actually inventing things.
The character they show on a regular basis doesn't have that capability.
He did when Stan Lee wrote the comic, and often thereafter. Not only did Peter invent the webfluid, the webfluid cartridges, the webshooters, the spider-tracers, the tracer launcher, the tracer tracking device, the improved tracers that he could track directly with his spider-sense, and other paraphernalia, but he was often shown to whip up a special chemical formula or gadget when needed to defeat a bad guy -- such as the formula he whipped up that let him dissolve the Rhino's armor in their first encounter. Later on, in the early '80s comics, I believe, he came up with improved webshooters that were plastic rather than metal (to get around metal detectors) and had LED indicators to let him know when his web cartridges were running low.
(And then, of course, there was the time when he designed and built the Spider-Mobile... but we don't want to get into that.)
It's always been an intrinsic part of Spidey's character that he solves problems with his intelligence and creativity, not just with muscle and superpowers. That's one of my favorite things about the character, and one of the things that makes him a great role model for younger audiences -- a hero who shows the power of science and imagination.