Thank you for all your kind answers! This is really thought-provoking. You know, when we first found it, we even thought it might be a doorbell mechanism?
It's been a while since I've seen the actual object (I took these photos last year when the item had just been discovered), so I'll have to ask my sensei to show it to me again once she returns from Japan.
In the meantime, I can say that there was some patina on the interior of the object, but aside from that, no traces of what it originally contained as far as I could see. Sensei had already given it a thorough cleaning, so, not too many clues there. Still, the next time I see it, I'll take a closer look inside.
Also, I think I was wrong about the age of the house. Upon reflection, I think it is possibly early Edwardian, not Victorian. So that takes us a few years forward.
The stable idea is an interesting one; however, I suspect my teacher is unlikely to know too much about the history of the house (she's a relatively recent owner, and doesn't speak a lot of English). But I have a friend who's something of a local historian, so I may be able to find out, at least, whether there were stables on that street.
Again, I haven't seen this object and examined its mechanism for a while. If I remember rightly, when the item is suspended, the lid of the "lamp" or "pot" is raised slightly due to the chain being taut - but even then, it's a relatively loose tautness, not at all what I expected!
I love the idea of this being a small lamp, but would be reluctant to put in oil and attempt to light it without really knowing what I'm doing. (The wick idea might be good

)
The incense-burner idea is interesting, but - one thing I forgot to mention is the size of the piece. The "pot" itself is only about 3 inches across, measuring from the end of the spout to the back. The sole incense I can imagine being burned in such a small container would be incense cones, and if I recall correctly, incense cones are a fairly new phenomenon - dating perhaps from the 1950s or 1960s. Of course, this item could be newer than I think. It's hard to know!
Actually, I'd be more than prepared to show this to an antiques specialist, purely for purposes of identification, but when you don't know anything about its original use, this becomes surprisingly challenging. The one local antiques dealer to whom I showed it drew a complete blank.
P.S. Correction: I just checked Kiyoko Morita's
The Book of Incense, and she says that Hata Rokubei of Shoyeido (Japanese incense company) "tried to be as innovative as possible; he created new kinds of incense (including the cone shape) and introduced Japanese koh [incense] at the World Exposition held in Chicaco in 1894. Only four years later, Shoyeido began exporting sets of incense and burners to America..." So it sounds like those cones have been around a lot longer than I thought!