Cameron, thanks for the great questions! Answers:
Lists are saved in the app server. The user gets a cookie in their browser that gives access to the list. So, yes, it does persist through multiple sessions in the same browser for at least 90 days from last access.
Yes, when a user logs into the shop, the list will be tied to their customer account.
Otherwise, a user can save the list to their email without using an account — the app emails them a secret link to recover the list any time in the future.