Multiple Checklists on an Work Items
Creating Multiple Local Checklists on a Work Item
You can create multiple local checklists, each with its own name and progress bar, on a single work item.

To create additional checklists on an work item:
Navigate to Jira settings > Apps and select Global Settings under the Checklists for Jira heading.
Ensure the Allow multiple checklists per work item toggle is on.
Go to the Jira work item and click on the + button on the checklist panel to add another checklist

By default, the additional checklists will be named Checklist #2, Checklist #3, etc. Clicking on the name will open the field so you can rename the list. Once additional checklists have been added you can also rename the default checklist.
When multiple checklists are present on a work item, you can choose vertical or tab view.
Limits of Checklists for Jira Free
Checklists for Jira FREE has a limit to the total number of checklist items per work item (as explained here. The limit applies to the total number of items in the work item, regardless of how the items are distributed across multiple checklists.
Multiple Checklists and the Checklist Text Field
When multiple checklists and Sync checklists to Custom Fields are enabled at the same time, the Checklist Text field will contain all of the local checklists on the work item.
Hiding the "Add checklist" Button
To hide the Add checklist button, disable the Allow multiple checklists per work item global setting.