Yes, you can do that for email and newsletter templates.
Taking the email templates as an example, in Communicate menu > Emails > you use the button "EDIT TEMPLATES" on the right side > then on the "Email Templates" page use the button "MANAGE CATEGORIES" to create, rename or delete template categories. A special case is the "Main" category that comes with the system and that cannot be renamed or deleted. You can rename and delete any other categories that you create yourself.
Then, when you create or edit email templates you can (re)assign them to which ever category you want, selecting it from the "Category" drop-down list at the top of the email template creation (or edit) page.
The process is the similar for organizing your newsletter templates into categories, via the path: Communicate menu > Newsletters > button "EDIT TEMPLATES" > button "MANAGE CATEGORIES".