Is a shopper's compare list saved to their account?

No. The compare list is saved in the shopper's browser on that device, not in their account. It stays there when they come back on the same browser, but it doesn't follow them to another phone or computer, and clearing browser data empties it.

Shoppers open their list on your store's compare page (/compare). If they switch devices, they just add the products again with the Compare button.