Furniture, frames for houses, and so on, are typically considered carpentry. Etymologically it comes from a Latin word meaning a wagon maker.
In contemporary English, at least in North America, it's mostly used to describe those involved in the construction of things like wooden buildings, particularly the structural components like frames and roofs. But it hasn't completely supplanted the broader sense.
In building terms we distinguish a first and second fix carpenter. First fix is mostly structural and second fix is the rest. If your hair is wet its probably first fix - roof trusses and the like. Second fix will include things like skirting boards. It's not a hard and fast rule and will probably involve the same people. These trades are known as "chippies" (sing. chippy).
We also have cabinetry (your furniture smith - nice one) and other wood working trades. They are all called carpenters. A chippy is a carpenter and I think this a rare case where the nickname is more common than the real name. Civilians will often call a chippy a "builder". A cabinet maker is never known as a builder or a chippy.
The chippy thing is because a very small piece of wood is a "chip".
I think I know why you are a bit confused about the word carpenter. In the US and elsewhere, houses are mostly wooden. When a house is nearly all wood, then the word carpenter should mean builder. In the UK and elsewhere houses are built of sticks and bricks, mostly bricks.
That said, I've never seen it called the "fish and chippery", but it wouldn't surprise me to see that used.
They just happened to be made out of wood and the people who made them were apparently willing to apply those skills to more than wagons.
In Latin-America we use a very similar word: "carpintero" with the same origin. But the houses in the region are commonly made of concrete brick, so we don't associate "carpenter" with "house builder".