This is resolved in 7.96 patch 13, released in September - October 2018 and applied to databases in Blackbaud Hosting Services. If your database is installed locally, download and install the latest version and patch, if applicable. 

An alternative solution in previous versions is to select the option to allow users to the Edit Constituent ID in the security group.

1. Create a security group in Admin > Security and unmark the Edit Constituent ID option in Records > Constituent > Miscellaneous Constituent Options
2. Login as a user who is a member of that security group, open a constituent and on the Relationship tab open a relationship for an organization that is a constituent in Raiser's Edge. The ID field to the right of the Organization Name field is blank.
3. Use the Arrow on the toolbar to open the constituent record for the organization and the ID shows.
4. Due to this, the ID is also blank when the organization relationship is included in export via Relationships > Organizations > ID.  Can include Is Constituent? in the export and this is populated with Yes in the exported file