Hello there Iain Elder, welcome to CTH!
An exe file does not need to have a file type in Windows since it is an executable file. It should be a program that something else is opened with. Windows inherently knows what to do with an executable (*.exe, *.com, *.bat) so there is no need to set any "open with" properties for executables.
Sounds like you might have some spyware or even a virus. You should head on over to the "
Cyber Safety" forum here and have some of the folks there help you out before trying anything else.