would certainly be a lot less confusing if the api was actually hosted on the same domain.

as-in: news.ycombinator.com/api

or something...

When I search for "news.ycombinator api" I get a bunch of irrelevant or very outdated links...

https://www.google.com/#q=news.ycombinator+api

I agree, but it's good marketing. It associates the act of searching with the hostname 'algolia', which seems to be in their interest (being that their tagline is "Hosted cloud search as a service | AlgoliaSearch").