Is it built-in in Sphinx?
Yes. Sphinx is not built-in, however. The search widget is part of sphinx. What context did you mean by "built-in"?
On the page iteself: http://docs.python.org/about.html
The Sphinx search engine is built in Javascript. It uses JQuery and a (sometimes very big) javascript file containing the search terms.
It look like Sphinx contains own search engine for English language. See http://sphinx.pocoo.org/_static/searchtools.js and searchindex.js/.json (see Sphinx docs index 36Kb, Python docs index 857Kb, and Grok docs 37Kb).
Index is being precomputed when docs are generated.
When one searches, static page is being loaded and then _static/searchtools.js extract search terms from query string, normalizes (case, stemming, etc.) them and looks up in searchindex.js as it is being loaded.
First search attempt takes rather long time, consecutive are much faster as index is cached in your browser.