Database and Web Connections should be shared via FME Flow in the same way as Deployment Parameters.
Doing so would not only be more intuitive, but it would provide more granular security since each user or role can be granted access to only the connections appropriate for that user or role.
The process described below is unintuitive because is not consistent with Deployment Parameters, and it would be tedious to set up multiple “Shared Database” of connections each with different permissions.