The download file contains the same fields that the Teacher can see on the Participants list. This always includes the first and last name, plus any other fields you have included in the setting Site admin > Users > Permissions > User policies :: Show user identity.
To prevent this, you would need to actually change the user's names in their profile.
You can limit certain fields of the user profile from view in the setting Site admin > Users > Permissions > User policies :: Hide user fields, but this does not apply to Teachers by default. To prevent Teachers from viewing the full profile of the users in their courses, you would need to take away (prohibit) the capability "View user profiles moodle/user:viewdetails" in the default Teacher role.
To prevent this, you would need to actually change the user's names in their profile.
You can limit certain fields of the user profile from view in the setting Site admin > Users > Permissions > User policies :: Hide user fields, but this does not apply to Teachers by default. To prevent Teachers from viewing the full profile of the users in their courses, you would need to take away (prohibit) the capability "View user profiles moodle/user:viewdetails" in the default Teacher role.
The only way to pseudo-anonymize a user is to change all the data in their account and user profile. You can keep a "key" to users by creating a custom user profile field for that key if you want. Then hide that field so that no regular users can see it. However, admins can always see all user fields.