What permission needs to be added to see the Security folder in Azure SQL? I have the Security folder, and I am trying to setup other Users with the same permissions that I have.

My User is not the Administrator. Though I was granted all of my permissions (which is about everything) by the Administrator. But this was done months ago and the Administrator (just a dude) does not remember what permissions they gave me.
As a minimum you need to be a member of loginmanager role to see Security folder.

Ref:https://docs.microsoft.com/en-us/azure/sql-database/sql-database-manage-logins


  Login managers
  
  The other administrative role is the login manager role. Members of
  this role can create new logins in the master database. If you wish,
  you can complete the same steps (create a login and user, and add a
  user to the loginmanager role) to enable a user to create new logins
  in the master. Usually logins are not necessary as Microsoft
  recommends using contained database users, which authenticate at the
  database-level instead of using users based on logins. For more
  information, see Contained Database Users - Making Your Database
  Portable.


Creating a login:

    USE [master]
    GO

   CREATE LOGIN [testPerm2] WITH 
   PASSWORD=N'strong PW'


Creating a user in master with no other privilege.

USE [master]
GO
CREATE USER testPerm2
    FOR LOGIN testPerm2
    WITH DEFAULT_SCHEMA = dbo
GO


Now connect using SSMS and this is what you see (Notice no security).



Adding testPerm2 user to loginmanager role.

EXEC sp_addrolemember N'loginmanager', N'testPerm2'
GO


Again connect with SSMS or refresh your previous connection and now testPerm2 user can see security tab.