Does the iOS App Store share information with developers that can identify specific users of their apps?
No, developers on the iOS App Store do not receive information from the App Store that uniquely identifies users of apps. That includes both free and paid apps. The developer also do not receive a list of device names of devices where their app was installed or purchased.
The developer can access only aggregate information, such as the number of purchases made by customers in a specific country.
Note that the developer might have other means of getting information about their users. For example by including a login system in their app, letting the user link in Facebook accounts, and similar. However that would be something that is separate from Apple's App Store.