What makes you think they’re PDF? InDesign exports PDF…it can’t save them.
A saved file acts like it is an indd file packaged in a PDF box.
When I finish a file and am ready to save it, the "save as type" looks right - "InDesign CC 2014 document." After saving the file, it displays on the desktop and in Windows Explorer with the Adobe Acrobat icon and "Adobe Acrobat document" as the file type. Then, when I click on the file to open it, Acrobat Reader tries but fails to open the file with a message that it is "not a supported file type." I can open the file from within InDesign, but I would many times much prefer to open indd files simply by clicking on the file name on the desktop or in Windows Explorer.
I use Mac OS, and I'm not very familiar with Windows, so the terminology I will use will probably be wrong, but I think the concept is the same on either platform.
On a Mac, I would right-click on the file and choose "Get Info" from the contextual menu. A window will pop up that looks like this:
I have circled the portion of the window that tells the OS what program to use to open the file. If you have a similar window in Windows OS, change the type from Acrobat to InDesign. Also, it used to be that all Windows files displayed a file-name extension (.pdf, .indd, .tiff, etc.), that would tell the OS what program to use to open the file. Macs in the old days didn't need that, because they used a different system (the resource fork). Current versions of Mac OS all have a file-name extension, but the OS allows the user to either show or hide that part of the file name when viewing files in the OS. If I take the same file I used in the above example and change it's file-name extension, I get this:
…and when I double-click on it, I get this:
…which I think is what you are getting. If I open it from within InDesign, it opens, but it won't open in Acrobat, because it isn't the right kind of file for that. If for some reason your file name doesn't have the right extension, you could also try changing it. That works in Mac OS, but I can't say which method will get this working for you. Also, if you use the first method, you can (in Mac OS), have the OS set the default program for any file of this type to open in a particular program. This is what you might do if you wanted all screenshots to open in Photoshop, as opposed to what ever program is the Windows default for viewing non-specific images (in Mac OS, it's a program called Preview).
Your File Association is, indeed, damaged. If it worked correctly at one time it might be worth trying Windows System Restore, but if not you really should reinstall.