Related Content By Taxonomy Term and fields
You don't need to duplicate Taxonomy Vocabulary, one is enough. 

You do need to have two Fields referencing the same Taxonomy Vocabulary, one for Department categorizing it, and the other for Select through which you're controlling what the related content is.  

This is still a case of showing sibling nodes but through criteria of matching one field to a different field that just happens to references the same taxonomy. 


Create Views listing your Organization nodes in a Block Display
Add Filter Select, set it to your vocabulary and then choose Is not empty (NOT NULL)
Add a Contextual filter for the Field Department and configure it:
 When the filter value is NOT available
Provide default value
Taxonomy term ID from URL
Load default filter from node page, that's good for related taxonomy blocks
Filter to items that share any term 
When the filter value IS available or a default is provided
Specify validation criteria
Taxonomy term ID
[your vocabulary]
One or more IDs separated by , or +
Place your block in a region shown only on Organizations, you will see nodes listed that are tagged with the term that is in the current node Select field. 


This solution relies on the concept that you won't have the same terms in Select Field that you have in the Department field.