It's right there in the definition...
A trope in that context is a narrational device. It most likely references the time when Florence Nightingale was nursing, the height of English Romanticism, as well as the historical person. It's not unusual for patients to fall for their caregivers, but in reality the specific trope is tied to wartime medicine and often depicts characters that are either mortally wounded or severely injured. In that sense it would be used to convey loneliness and the austerity of the human condition.
All of this is to say you're not experiencing the Florence Nightingale effect.