Does your wallpaper follow your account
Signed in, your choice is saved to your account and follows you.
The short answer
It depends on whether you are signed in. Signed out, your wallpaper choice is kept in the browser you chose it in. Signed in, it is kept on your account and follows you to other devices. A choice you made before signing up carries over to your account the first time.
Signed out: the browser keeps it
Wallpapers need no account, so the site has to remember your pick somewhere. Signed out, that somewhere is the browser. The web platform gives every site its own storage area: the WHATWG HTML standard says that in web storage each site has its own separate storage area, held on the client side.
The practical result is that the choice belongs to that browser on that device. Open the site on your phone, or in a different browser on the same computer, and the default background is what you see. Clearing site data in the browser would also clear the choice.
Signed in: it follows your account
Once you are signed in, the choice is kept on your account instead. Sign in on another device and the same wallpaper is there, with nothing to set up.
That is the whole reason to sign in for this feature. It costs nothing, and it removes the dependence on one browser holding on to your pick.
Choosing before you sign up
You do not lose a wallpaper by registering late. If you picked one before you had an account, it carries over to your account the first time you sign up or sign in. After that, the account copy is the one that travels.
Getting started walks through the order that makes sense for a new visitor, and picking a background first is a perfectly good opener.
If your wallpaper has not followed you
If a device shows the default background after you signed in elsewhere, first confirm you are signed in to the same account on both. Signed out, the choice lives in that browser only, so it will not appear.
If you are signed in and it still differs, choose the wallpaper again on that device. The choice is saved to your account either way.
A short worked case
Suppose you pick a wallpaper on your laptop before you have an account, then sign up the next day. The first time you register or sign in, that pick carries over, so the laptop looks the same as before. Sign in on your phone later and the same wallpaper follows, because it now lives on your account.
Now suppose you never sign up. The pick stays on the laptop’s browser and nowhere else, which is fine if that is the only place you use the site.
What follows you and what does not
The wallpaper choice is what travels with your account. The site does not publish a longer list of synced settings for wallpapers, so treat the choice itself as the thing to count on and check anything else on the screen.
Changing or resetting it
Tap the active card again, or use Reset to default background, to go back to the standard desktop. The reset follows the same rule as the choice: it is kept in the browser when signed out and on your account when signed in.
For how to pick and apply a wallpaper in the first place, see how to download and set wallpapers.