Buying OpenRouter has nothing to do with APIs.
Stripe wants to be in the business of collecting a % on the economy at large. This is why people buy Visa or Mastercard stock. It is not because of the tech but because it's effectively a tracker for economic activity (you could also argue rent-seeking over the economy but whatever).
Stripe can then sell potential investors on their stock being basically a tracker on the AI economy. Now you get access to private AI companies growth as well! IPO time!