If you go into Modules>Contact Management, you can add your custom data fields in the Contact management - Contact class. What I do is typically add a category for those custom fields and set visibility on that category based on the current site name. Then since you have a multi-tenancy setup, only that particular site will see them. Just make sure those fields are set as not required. You can create alternate forms with required fields if you need.
Contacts are specific to a site.