It is possible to have the paid version of Acrobat and the free Reader installed. If you are on the same computer, it is possible that if you had the paid version (that could make edits) that it is still installed.
When you install new software, it likes to say "Shall I be the default?" So installing free Reader would change the icons. And double-clicking the file would open in the new software.
If my guess is correct, then all you need to do is change the association for .pdf to the other version of Acrobat where you can edit. Or open that version, then choose File > Open. Or get a newer version of Acrobat Standand or Pro.
Check to see if you have another version of Acrobat installed. This is just a hunch, based on what you've said. Let us know if it's there, and we can further advise.