CiviEvent multiple participant registration does not save all field values to contact
I think I have found the solution to my problem. I'll post this here in case someone ever stumbles on the same problem.

When configuring the profile that you would like to use for the event registration form, be sure to not set the e-mail type (category) to "Billing". If set, the email will not be saved to the contact (no idea why).

All other e-mail categories seem to work fine (e.g. "Primary" or in my case "Main").