Option 'Add to Starred' Not Available for Jira Company Managed Projects
Platform Notice: Cloud Only - This article only applies to Atlassian products on the cloud platform.
Summary
As shown in the screenshot below, the Jira company-managed projects do not have the option "Add to starred."

Environment
Jira Cloud
Diagnosis
Users unable to see 'Star' option for newly created company-managed Jira projects
Cause
Users must be granted "Browse Project" permissions in the relevant permission schemes linked to the project in order to be able to star the project.
Solution
To add users, groups, or roles to the permission scheme and assign them "Browse Project" permissions, follow these steps:
Select Settings (Cog Icon) > Issues.
Select Permission Schemes to open the Permission Schemes page, which displays a list of all permission schemes in your Jira system and the projects that use each scheme.
Locate the permission scheme you would like to update, and select Permissions in the Actions column to view the scheme.
For the Browse project permission available, select Update to grant the permission to a user, group, or role. This displays the Grant permission dialog.
In the Grant permission dialog, select who to grant the permission to and click the Grant button. You can grant permissions to:
Individual users
Groups
Project roles
Issue roles such as Reporter, Project Lead, and Current Assignee.
Please follow the below documentation to understand more about the Jira project permissions Manage project permissions and Permissions for company-managed projects
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