I think they just mean that it's supposed to be an easy, versatile API for photo integration into whatever web app you're working on, like how Twilio is an API for phone integration.
You know that product {X} has arrived when it's assumed to be so mainstream that is's being used to describe functionality {Y} in sentences like "it's like {X} for {Y}".
The Twilio reference is more a nod to how effective Twilio has been at making voice accessible to developers.
It's more an analogy and definitely not the ideal, most precise way to describe what we do.
We're much more likely to consider ourselves akin to Heroku as we offer infrastructure and connections to services that make deploying and running super simple. Not to say that Twilio doesn't accomplish the same for tons of folks.