In default odoo we don't have functionality to select and use same stages on multiple project. Advance Project Stage Organiser odoo app helps user to set only selected stages for project, user can select multiple stages and only this selected stages will appear to related project, if user removes stages from project, project will also removed from stages, User can also share same stages on multiple projects.

Project Stages

While creating new project user can see that all stages will added by default to project.

On project form view user can see "Tasks Stages" fields and can select and set default stages for project.

Now user can select same stages on multiple project, While creating project user can choose/select required stages on that project and share same stages on multiple projects.

User can see that only selected stages will appear to projects, User can also select same stages for multiple projects.

On adding or removing "Tasks Stages" on project form view, "Projects" on stages also updated.