Yes, you can see your total Records and Files Count on the Mass File Download Tab. You can see the files on any tab inside the Mass File Download tab (Related Files, Static Resources, Documents, Files).

Go To Mass File Download tab, 

  1. Select an Object.
  2. Select a ListView (If Show List Views toggle is ON)/ Search for a record (If Show List Views toggle is OFF)
  3. Select Filter for Related Files and Attachments (Optional).
  4. Click "Search".

You will be displayed with the Files Count with Parenthesis in blue color. 

eg. Existing Related Files & Attachments (3433).

Under that, you will see the number of records having those many files.

Showing (3433) Files & Attachments of (14) Account records.