Is it possible to assign an anonymous user's flags to a new user on registration?
I am not aware of any module that already does this with flags. But you can take a look at Anonymous publishing. Its "Lazy Registration" mode will assign any node created by the anonymous user to a newly created user if that anonymous user later registers an account.

The method used is to require the anonymous user to submit and confirm an email address when creating a node. When somebody with that email register, the nodes tagged with that email address can be assigned to that newly created user.

It should not be difficult to use the same method to assign for flags/requests created by the anonymous user to a newly created user.

If you can allow for some errors, you can use something less intrusive than a confirmed email address to link the anonymous user to the newly created user, such as a persistent cookie (will fail when the cookie expires or the user moves to a different computer), or an IP-address (will fail if the user moves to a different computer, if two different users users use the same computer, or if a dynamic IP-address gets reassigned.)