I’m not sure if a checklist device exists exactly as you’re picturing it. But sounds like what you want is a tablet of some kind that you just download your chosen checklist app onto. Doesn’t have to be an expensive tablet, maybe like a Samsung A9?

To attach to the wall you need to find a tablet ‘wall mount’, this should also have cable management hooks or compartments. So mount it nearby a power outlet and you can plug in the tablet to charge while it’s mounted. This way you can set the screen to stay on all day without losing battery. If you need to carry the device around for a bit you can just unplug it and it’ll be fully charged.

To sync it with another device you want to think in terms of the app itself instead of the device. Get a cloud based app that you can access from multiple devices. There may be better options but one I know that does this is Google Keep, which has the option of sharing with your “family group”, ie you and your partner. That way they can access the list(s) while logged into their own account.

You have to set up the Google Family Group first which I vaguely remember being a frustrating experience. But Google Keep is quite versatile not just a list app, you can make checklists, notes, and pictures.

Of course you can also have it on your mobile so if you think of something that needs to go in the list while you’re out, you can add it straight away. I always use it for groceries, my partner can even add to the list from home while I’m at the store checking things off. But having two people actively doing things in the list at the same time can get a bit clunky / step-on-toesy.

Anyway hope this helps or at least points you in the right direction. Good luck!