The verb censer is obsolete and you won’t find it in day-to-day use. Only the past participle of censer → censé is commonly used.

From my experience as a native (Swiss) French speaker, censé faire quelque chose is as common as the English locution to be supposed to do something. There is nothing formal about it.

See Le Trésor de la langue française informatisé.

I hear its past participle used in negative constructions quite often when my French wife lets me know that I have done/am doing something that I shouldn’t:

T’es/étais même pas censé le faire !

Or else when I’ve done/am doing something incorrectly:

T’es/étais pas censé (le) faire comme ça !

Except to the extent that she usually says these things while she’s “formally” reading me the “Riot Act,” I’d agree with Pierre Arnaud that there’s nothing formal about it, neither in its negative nor its affirmative form.

As an aside, I’ll add that the false friends/cognates, sense and censé, at least in my two negative examples, help me to get what she probably really means:

You don’t have enough sense to know how to do it (correctly)!