This is a workspace role difference. Editing the private network hostname falls under "Modifying service settings," which is only available to the Admin role. Members and Deployers cannot modify service settings. You can check your role in the workspace People settings, and a workspace Admin can upgrade your role if needed.

I am an admin within our organization. I did not create this project, but also in a project i created i don't have this option. Someone who is member can do this.

Apologies for the incorrect assumption. The workspace permissions table does confirm that only Admins can modify service settings, so if you are an Admin and a Member can edit the private network hostname but you cannot, that sounds like a bug. Could you double-check your exact role under Workspace Settings > People and confirm both your role and your coworker's role? If you are indeed Admin and they are Member, we'd like to look into this further.