If your organization supports licensing for multiple jurisdictions, you will need to add the license to the animal's record from the Add License section on the profile tab of their record.
When you add a license this way you can edit the jurisdiction while adding the license information, then you can run either a Licenses Issued Quick Report, or a Custom Events License report to view the jurisdiction, issue date, expiration date, as well as many other related fields as well.
This video shows you how to edit the Issuer while adding a license from the profile tab on an animal record.
You can view the issuer/jurisdiction and license issue or expiration date in the Licenses Issued Quick Report:
You can also see this same information in a Custom Events License report: