Although all users can create their own projects, the Lab admin can determine whether the lab members are allowed to use their own projects for a booking.

If you have created your own project, under My Project - see image for reference - and you are not able to select it when creating a booking, it is most likely due to the admin not allowing users to use their own projects in the lab

Contact the lab admin and request they create a project for you, add you to an existing one, or enable their lab members to use their own projects. They can do this by going to the Project section in the lab, and switching the toggle to Yes - see below image for reference.