Rather than trying to find postmeta that matches (postmeta queries are typically very slow), it sounds like you would do better to work with Custom Post Types and Taxonomies.
“Organization” would be a custom post type.
Any people you may want to reference differently, like your examples of “Advisor” and “Team Member,” would work as a Taxonomy. So for example, you would create an “Advisor” taxonomy and a term for each individual Advisor. You would apply your “Advisor” taxonomy to the “Organization” post type, so you could assign however many advisors you want to each organization, and you could then more easily query for all Advisors who are assigned to multiple Organizations, listing the links to each Organization on the side as in your screenshot.
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