to fetch a name of projects where a specific user has an access to, you can use OpenStack CLI and a command:

openstack project list --user <user_id>

Is it possible without specifying a project?
Directly fetching all servers without specifying any project is not possible using the standard OpenStack API. This is because OpenStack requires a project context for most operations. The solution is to create a custom application or script that automates the process of retrieving data from multiple projects.

/Jan Wasilewski