How to network Organizations?
AFAIK there's no module which does this out-of-the-box for organic groups - the User Relationships module does this for users (https://www.drupal.org/project/user_relationships).

To achieve with groups, you could set up a similar system with flags and rules, i.e. someone can flag a group as a 'friend' or whatever, when they do, a rule sends a confirmation link to the friended group and they can then say they are friends with the other group (also using a flag). But I don't see why you'd do that as every person on the site is a user so you'd need to work out who has the ability to 'friend groups'.

A general observation - if you've looked around a lot for the functionality and it's not there, and it's fairly generic functionality (like 'friending') then you may be thinking in the wrong way about the problem and the problem may be solved by thinking about it different architecturally - I've been using Drupal for 8 years now and not had issues finding modules which fit to do the stuff I want to do.

HTH

Steve