I know this post is a couple months old, but the workflow I used is as follows. It assumes that your object IDs in Point 1 and Point 2 match. Point 1 has attachments. Point 2 does not.
1. Use enable attachments on Point 2.
2. Use append. Input dataset is the Point 1 Attachments Table. Target dataset is the Point 2 Attachments Table.
The append tool will copy all info from Point 1 attachments to Point 2 attachments. Then Point 2's relationship class (created by the enable attachments tool) should correctly pickup and associate the attachments, assuming your object IDs match.
Hope that helps.