If your tenant is currently configured as a Classic Organization + Flex Orgs tenant, to view FlexOrgs within the Organization Switcher and in Setup -> Admin -> FlexOrgs, a user must have a Role Document assigned via a Usergroup.
The assignment will determine which FlexOrgs are visible within the Organization Switcher. To provide access to all FlexOrgs for a Classic Organization + Flex Orgs tenant, you should assign the user to a Usergroup that assigns the Role Document at the root Organization level or Top Level Organizational Unit (TLOU).
You can determine if your tenant is configured as Classic + FlexOrgs by navigating to Setup -> Admin and verifying if the two options appear. If they do, your tenant is running with the TLOU set as a Classic Organization.