There's a dozen code documentation solutions out there. They could have adopted one that already handled Markdown and simply added extensions to it; instead they reinvented the wheel.
They sell an API product, of which documentation is an absolutely core, user-facing part of their product. It doesn’t make sense not to own something that differentiates your products value proposition.
They are making documentation for developers.
And in fact, having good documentation and developer tooling is a core part of their value proposition.
Making it easy to look at documentation and immediately implement something that works is a core part of the Stripe UX.
So documentation can be considered as part of their product offering.
In that way it makes sense for them for it to be driven by code.