You will need the Full Read Contributions permission to run giving reports.
Short answer: once your giving is split by organization, a giving report reads one organization at a time. The Organization field is there because adding two organizations' giving into one total would be wrong, since each organization is a separate legal entity with its own books.
Giving is split whenever your account has two or more active organizations. It also stays split after you go back to one organization, for as long as an archived organization still owns gifts, batches, recurring gifts, pledges or journal entries. When giving is not split, the field does not appear at all.
Which reports ask
While giving is split, these reports each cover one organization:
Batch Summary
Combined Contributions
Contribution Range
Lapsed Donors
The Organization field on these reports:
Is required when more than one organization is offered.
Shows the organization's name as fixed text when only one is offered.
Is hidden when giving is not split.
It offers the active organizations your campus selector reaches where you hold Full Read Contributions at one or more campuses. After those, it offers archived organizations that own gifts or batches, if you can read all of their giving.
Where two organizations can still appear together
Not every giving screen forces you to pick one organization. These can show more than one:
The Batches overview cards - their totals add across organizations unless you narrow them.
The recurring gifts overview - can include more than one organization.
The Contributions list - has an Organization column and shows no total, so nothing is added together.
Query Builder results - a total adds across organizations unless you narrow the query.
Tip: A figure that looks too high is usually two organizations added together, not a bug. Check for an Organization filter and narrow it to the organization you want.
Other things to know
Built-in giving reports and giving widgets read one organization at a time. If an organization is refused or not offered to you, the report does not quietly run on another one instead.
Batch Summary narrows by batch, not by each gift, so a batch always appears complete to whoever can see it.
Pledges count in their campaign's organization. A gift or pledge with no organization counts in the default organization.
Giving statements run one per organization for each donor, including donors who only have a pledge. How do I generate giving statements for multiple people? covers running them.
Designation Details has no Organization field. It reads one designation and is headed with that designation's organization.
A report's heading shows the organization of the one record the report is about, otherwise the one organization it covers, otherwise the default organization. Reports on an archived organization's books are headed with that organization's name.
Note: The AI Assistant works differently. Instead of narrowing to one organization, it must show an Organization column and split its totals by organization. See How do I save and use an AI report?