Just curious why the prior owner put this wire in the engine fan relay? Probably a dumb question but curious....

Maybe so he could bite it and electrocute himself with all the problems he was having ?? lol....It looks like its a simple loop...If so there might have been a problem and a relay bypass was needed...

Does the fan turn on when you turn the key to RUN?

No it doesn't

Is the wire actually plugged into some contacts? Did you pull the relay to see what pins it bridges? You can always try removing the wire and see what doesn't work.

Some folks address overheating by bridging the fan relay, usually with a switch, to get more control over fan.