You can picture chancery as one room: the room where an institution keeps its official paper. Clerks at desks, documents being copied out, a seal pressed into wax, shelves of records that nobody throws away. That room is the whole word. When the person who ran the room began hearing people's complaints as well, the room grew into a court. When other kinds of institution needed the same room, the name went with it — an embassy has one, a diocese has one. Whatever the building, the picture underneath is the same: this is where the official papers are made, sealed and kept.
From the word’s history
Chancery comes from Middle English chancerie, a shortened form of chancellerie — the chancellery, the chancellor's office. So the office is not a later application of the word; the office is what the word originally named.
chancery 來自中古英文 chancerie，是 chancellerie（chancellery，大臣的辦公處）的縮短形。所以「辦公室」不是後來才有的引申，它本來就是這個字在講的東西。
Wiktionary etymology supplied in this packet (source_ref kaikki:3f6f69a190036ae4:919608078): From Middle English chancerie, a reduced form of chancellerie. See chancellery.
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