Hi Jonas,
Thanks for your mention, my word would not be correct, "unique ID" should be more proper to describe that long string.
Could you give more details about "a recognisable cookie in their browser"?
For Chrome,
you can find the automatically created cookies of subscription center form in
Settings > Privacy and security > Site Settings > Cookies and site data > See all cookies and site data

However, there are limited infomation for us due to security restriction.

So cookies should have be saved in browser once user opened subscription center link and saw prefilled data of him/her.
For issue on your mobile phone, that's bit strange,
I tested on Outlook app on iOS and the default browser is Safari, I didn't encounter such issue.
would you had opened the link in incognito/private mode? because cookie might be affected in this mode.
(or there was delay when you first time opened the link)
In addition:
1. I used embedded form on external website to test.
2. Prefilled data will change if I open the link from different mailbox, so saved cookie shall change at same time.
You could share your setup settings if you had any doubt.
Regards,
Clofly