Right, so here's how I understand it: the first Switch that you link your Nintendo Account to is deemed your 'primary'. This isn't a choice like on PS4 where you can activate/deactivate it manually, it's automatic the first time you do it. Therefore, I can guess from here that your Switch is your primary and your wife's Switch is hers.
If I'm doing it in my head right, then you can play New Horizons together from one copy of the game by:
a) buying it digitally (€41.13, cough cough)
b) having your primary account on your wife's Switch
c) having a second profile linked to your Nintendo Account on your Switch
You can have a user account linked to your Nintendo Account on multiple devices and download/play your games on all of them. The only difference is that on devices that aren't your primary, the Switch needs to connect to the internet first to see if the game is playable. When the Switch connects to the internet, it's checking the status of your primary device: if your primary device isn't showing that the user linked to your Nintendo Account is playing on it, then you can play. If it is, then play on the primary Switch will be suspended and play on the other will begin. I'm not sure if you have to be playing the same game or ANY game for the suspension to happen but, basically, you can't be logged into two places at the same time. However, software downloaded through your Nintendo Account on your primary Switch can be played by ANY user registered to the Switch… this doesn't count as you being logged in.
Transferring the primary from one Switch to another is NOT the same process as transferring your user profile from one Switch to another. This deregistering process doesn't move your user data or saves over as the proper transfer process does. But if you've got NSO and you make sure all your saves are backed up onto it, that's not an issue (and also, there's no reason to put your saves on the other Switch as you'll see below).
Now, to the point: I think this is doable. For New Horizons multiplayer, there's the obvious 'Both play on one Switch, live on the same island' option, but I'm guessing you want your own individual islands. For that, you need to play on two different Switches, which you have. In my head then, the following SHOULD work (though I can't confirm until it's out and we can test it!):
- Follow the instructions on this page to deregister your primary console. Your user profile and Nintendo Account will still be on your Switch, along with all your saves (but make sure they're backed up to NSO cloud, just to be safe)
- Create a new user profile (it can have the same name as your original profile) on your wife's Switch and link your Nintendo Account to it when prompted
- Use your wife's Switch to log into Nintendo eShop with this new user profile. This then registers your wife's Switch as your primary console.
- Buy and download New Horizons from eShop through your account onto your wife's Switch. Since it's your primary Switch, she can play the game through her profile without the system running any checks.
(At this point, you can also use your cloud saves to download your saves to her Switch, but I don't know why you'd want to…)
- Go back to your Switch, go to the end of the HOME screen and click the circle with boxes in to see your whole library. At the bottom, select Redownload Software and then download New Horizons onto your Switch. Now you can also play New Horizons, but only when connected to the internet (as the Switch needs to check first if you're allowed to play… if the check fails, it'll fail your attempt to load the game)
Because your Switch is no longer your primary, there's another catch on top of the 'checking the internet' thing: only your profile can play games you own. If a game is multiplayer, others can play with you locally providing the game is first booted up through your profile. However, no games you own can be loaded directly by another user profile on that Switch.
Sounds complicated, but it's not. I'd be interested to see if this works… it should!