Can members sign-up for two memberships from same organization?
Yes, introducing a second "dummy" organisation is the way to go. This is described in more detail in the user documentation:


  Some organisations may want people to have two or more memberships of the same organisation that run concurrently. For example, an organisation focused on child health might want to offer a membership for parents that includes a parenting magazine and a membership for health professionals that includes a peer-reviewed journal and discounts at training events. Parents who are health professionals may want both memberships. A challenge to supporting this use case is that CiviCRM's Membership Up-Sell feature expects a contact to have only one active membership with a single organization at any given time. A 'workaround' for this is to create 'dummy' organisations for each of the possible concurrent memberships. For the situation just described, we would need to create an extra organisation for health professionals. Note that you don't have to expose the dummy organisation to your members on the website; it is for administrative purposes only.