When a user launches H5P.com through the LMS (LTI), what is identifying the user is something called user_id. This will not change even if the email sent by the LMS changes. I.e: the user will still be logged in to the user account they had for the "old" email address, and the user will see the same content.
If you want the email address on h5p.com to be the same as the LMS is sending through LTI, you can either manually edit the user accounts on h5p.com, or set up the "LMS connection" to always update the email from the LMS (as described below)
LTI 1.3: Automatically update the email address
For LTI 1.3, navigate to Manage Organization->Connect LMS, and click the Connection info button on the relevant LMS Connection. You will then get a dialog, where you'll find the checkbox highlighted below. Enable this, and click Save.
LTI 1.1: Automatically update the email address
For LTI 1.1, navigate to Manage Organization->Connect LMS, and locate the relevant LMS Connection. Here you'll find the checkbox highlighted below. Enable this.