I’ve had similar problems in the past with other softwares. The only way I managed solving it was by editing the registry.
Search it for “.unity” without the quotes and check “full word only” (not sure if it’s called that because my Windows is in French) and delete the .unity results you find; there might be more than one. Save your registry before doing that. Then restart your computer and reassign .unity to the correct value.
Your screenshot doesn’t show that you changed the file association. What happens when you set Unity and not the Installer, using Change Program? You’ll need to be an administrator.