Hey, I just tried to edit my profile page without signing in, and it worked! It's dosen't supposed to be possible, because everyone can do it.
I had also tried to change my image without signing in, and it's didn't let me.
When I signed in, however, It's seems that I can edit and change the profile page of other users too, including the image (I can't be sure because I didn't want to press the save button, but you can try it yourself on my account).
Hi! So yes, for some reason FANDOM allows unregistered users and users to edit others' profiles. However, profile pictures and user information (The part beside your edit count) can only be edited by the user themselves.
If you'd like, I can protect your profile for you so only you are able to access it :)