Tickets are not assigned to Organizations. They are put in Department or Team containers.
If a ticket belongs to an organization it is because they are owned by a user in the Organization. To set that up:
Go to Agent panel -> Users -> Organizations
And click on the Organization you want to edit.
Then click on the name again.
Then click on the Settings tab
You can setup Auto Add Members from an email domain.
This means any Emails from that domain would go to that Organization.
If your Organizations are actually Departments within your Company then you likely cant do this and get it to work since they all have the same email domain.
If your really just trying to get the tickets to show up in specific departments Ticket Filters would let you assign tickets to a specific Department then this would do the trick (not hypothetically since I have not done it myself):
To make a filter that should find your variable you would:
Go to Admin panel -> Manage -> Filters
Add New Filter
in Matching Criteria select Ticket / Issue Details (which would be the email body)
Select Matches Regex
And in the text field enter the text that you want to match.
Then under Filter actions select Set Department.
Then select the Department that you want to tickets to go to.
Either way please let me know if you need more assistance with this.
