The modern usages all derive from the older sense of the word - a "little room" (or cabin), especially one in which special things are displayed or special events take place.
It is a short step from here to find the word applying to the furniture within a room, or to the people who meet there.
In the Shakespeare canon, the word occurs only three times - twice in the long poems - and in each case it is the oldest meaning which is retained.
The sense of lodging or dwelling-place is found in The Rape of Lucrece (442), which talks of a "quiet cabinet" and Venus and Adonis (854), where the lark "from his moist cabinet mounts up on high" (in other words, from his nest).
In King Edward III (II.i.62), the king tells his secretary Lodowick to make an arbour "our counsel house or cabinet". Here the sense is one of intimacy - a private meeting-place - for his intention is to compose a love letter there.