Using multiple organisation means you will be separating members between organizations. Sounds like it will get messy trying to assign owners while at the same time not double-counting across organizations.
Using one organization and different boards based on projects instead of teams could be an approach to use. If there is a case in which one team always ends up doing the same types of projects then they are naturally assigned to it. Otherwise, where there is a manager that oversee more than one project/team, you will be able to add to multiple boards.
The moment you know that there will be a large amount of crossover between teams should indicate when you should use one organisation.
In terms of whether too many boards will fill the main organization area, you can use the "Visibility: Who can see the board" to adjust that. Thus team members only see what projects they are a part of.