Purpose
The Jira Issues and Jira Comments components do not have the option to switch the Jira connection via the dropdown option as shown in this documentation.
Answer
Only default connections will be shown in the Lightning Aura and VisualForce components. The connection shown can be confirmed by hovering over the 
tool tip.
To change the default connection, read How do I connect more than one Jira connection in my Salesforce environment?
Related content
Is there any difference in setting up the integration in Jira for the multi-site feature?
How do I connect more than one Jira connection in my Salesforce environment?
How to view, create and associate Jira Issues from Salesforce by switching the Jira connection?
Are attachments sync configurations configured individually with multiple Jira site integration?
What are the limitations of the multiple Jira site integration?