The language drift is absolutely confusing here but "establishment" as "organization" seems to be quite correct. Establishment is also used (even to this day) as a vague word for location in a utilitarian sense, "Business's establishments" is one surviving usage of the term that comes close to that archaic usage - it is a tool or asset in support of the operation of a thing, a thing established in support of the thing.

Lastly I wanted to mention the parallel term you draw here because it's also a drifted phrase

> to make official

Is actually a highly accurate synonym for establishment, but only in archaic word usage, official (the adjective) is actually just a corruption of official a noun, it's sort of like "the influencer lifestyle" or "that's an influencer hat". Technically all "off-*" words originally related to religion, but that's gone by the wayside quite a bit by now.