I know this sounds crazy, but hear me out. I came up with this idea last week, and it has been driving me insane since. Captain America: Civil War takes place in 2016, and Agents of S.H.I.E.L.D season 2/3 takes place 2015/2016. In Civil War, Peter says he had his powers for 6 months. At the end of season 2 in Agents of S.H.I.E.L.D, it is revealed that Jiaying and other inhumans released Terrigen Crystals to the public in the form of fish oil, right around the time Peter said he got his powers. If Peter took some of the infected fish oil, or he ate fish, like Yo-Yo, and he had some inhuman genes, then he could very well be an inhuman. In Agents of S.H.I.E.L.D, it is mentioned by Lincoln Campbell that inhumans have a purpose and their powers are based on what the world needs. Peter became Spider-Man right around the same time the Avengers needed him, hence his purpose. The spider bite could have been completely coincidental, or maybe it helped influence his powers somehow. Maybe he never figured out how exactly he got his powers. It's never explained how he got his powers. He could have gone through Terrigenesis. If he had, he would have been so confused because of his powers and the process of Terrigenesis. Maybe he was so scared about his powers, he never told anyone, including Tony Stark.
Let me know what you all think!