Note: you can only assign a collaborator to an organization project after that collaborator is first successfully connected to your organization.
There are two ways to assign a collaborator to a project: either from the project's detail screen, or from the collaborator's.
From the project
Open the project details by going to the Projects overview page, and then clicking the details-icon on the left of the project name. Then go to the tab Collaborators. This tab allows you to assign as many collaborators as you want to this project.
This way is especially useful when you create a new project.
From the collaborator
Open the collaborator details by going to the Collaborators page and clicking the icon on the left of the collaborator's name. then go to the tab Projects. This tab allows you to assign this collaborator to as many projects as you like!
This way is especially useful when you've connected a new collaborator.