Is there any chance to use an attached file (to an automated test case) to be copied to the test agent and open them in the test method? Or is it only a documentation attachment?
In TFS to help track a work item (test case here), we can attach a file with supplemental information.
For example, you can attach a screen image that illustrates a problem, a line of code in a text file, a log, an e-mail thread, or a product feature's specification. See Attach files
But in general we cannot and no needed to open the attachments directly on test agent machine.
Of course you can write method or scripts with the API to download the attachments and open them on test agent machine if the machine matches the requirements.
Reference REST API to Download an attachment and Downloading Attachments from TFS
If you mean EMBEDDED the files in unit tests, you can reference this article : FILES AS EMBEDDED RESOURCES IN UNIT TESTS