You can have exactly 1 SAP per Business Unit.
Assuming you bought 10 SAPs, you'd need 10 BUs to install them.
Private Domains indeed do not brand images and links.
Quoting an SF presentation on the subject:
Q: Can we put multiple SAPs on 1 account? A: No, a single MID can only support 1 SAP for branding purposes (i.e. the links and images can only point to ...