Netlify being related to Gatsby and Vercel being related to Next.js.
Not exactly. More that Netlify started as a no-devops-needed static website hosting platform, and gradually branched off into value props that naturally fit into that starting point.
Vercel (formerly ZEIT) basically did the same thing.
Since founding, both companies have raised an insane amount of VC money, making them both unicorns by valuation a few times over. Though imo it remains to be seen if this was actually a good investment. Neither company seems to have fully made good on their business model, which is quite similar in both cases: encourage and allow devs (or non devs) to do the absolute bare minimum of work in exchange for hosting/bandwidth/etc prices that many consider exorbitant.