If anything the opposite is true. You sign up for an email address and own that account. If the account is attacked then the consequences can be significant. Attack a mailbox just right and the user can lose access to it permanently.
IPs are allocated to you from a pool and your last IP will change as you move between devices and networks - home/school/work/cell connection and coffee shop for example.
I would be much more concerned about who you allow to see your users data, than their ability to unmask an IP address.