Zoning is typically for building. This would be closer to deed restrictions or “covenants, conditions and restrictions” (CCRs), which governs land use and transfer restrictions regardless of ownership.
Zoning actually addresses both construction and land use and often a lot of other things. Where I live, it would be absolutely routine to see something like short-term rental restrictions in the zoning code.
By the book, "Zoning is a legislative act dividing a jurisdiction's land into sections and regulating different land uses in each section in accordance with a zoning ordinance" [1]. So in theory, zoning rules deal with any kind of "land use", however the governing body chooses to approach the topic. In practice, you'll of course see zoning district rules that describe some of the more obvious "allowed uses" such as "dwellings, one-family, other than mobile homes" as well as construction/building characteristics like "detached homes", "townhomes", "multi-story buildings", etc. I think that is the kind of thing you're expecting of a zoning code, and it's definitely a big part.
But, zoning regulations get quite specific in dealing with uses too, and often these details have little or nothing to do with the buildings or construction. A major example is what kinds of business uses are allowed in a particular zone, for example: "Retail stores selling soft goods, clothing, leather goods, health aids, eye glasses, toys, jewelry, cosmetics, printed materials, glassware, home furnishings..."
It goes beyond characterizing the primary use of a parcel or structure. "Accessory" and "prohibited" uses are often even more specific, e.g. "entertainment (piano player, guitarist, small combos, dancing, etc.) in restaurants and movie theaters". You'll also see regulations for things like in-home occupations (hairstylists, massage therapists, etc), gardening, operation of vending machines, storage of construction materials and refuse, butchery and meat processing, etc.
Certain "performance" requirements are also sometimes listed, which usually deal with noise, light emission, and other nuisances, but these can get as oddly specific as this: "All surfaces shall be of a dust-free nature."
All examples listed are from the zoning code where I live [2].
Yep, it would not be exceptional to see short-term rental regulations in zoning codes in the US. I'm guessing in some states you'd have legal challenges if you required owner occupancy or even perhaps establishment of residency or domicile, but it's fairly universal in this country that you could write zoning laws dealing with transient occupancy and accessory or primary uses that resemble hotels, guesthouses, B&Bs, etc.
Yeah, it's a kind of zoning. But a quite strict one. And one that's not easily abused.
For instance, if you're tax based in Oslo, you would have to "move" to buy a family home in my city and use it as a cabin. And that move on paper has implications. Where you can vote, where your children can go to kindergarten or school, how you're taxed etc. So people don't do it.
https://localhousingsolutions.org/housing-policy-library/dee...
https://en.wikipedia.org/wiki/Covenant_(law)