No...
It's something else entirely.
Whether the stories about them is true or not is a mystery. The real name of Majnoon was Qays ibn al-Mulawwih. He was called "Majnoon" which means "a mad man", because his love for her was so much, he lost his mind. Now, there are differing reports about why they were not able to get married: some say it's because he was a prince and she was a slave, others say it's because she was a princess and he was a pauper and a nobody. The custom in those days was to only marry someone on the same social standing as yourself. There is more evidence to show that he was the prince and she was a slave and thus his family prevented him from marrying her, despite the fact that they were childhood friends.
When Layla passed away, Majnoon used to wander the streets of the neighbourhood wherein she lived. He used to say that he loved her so much, he even loved the dogs in the neighbourhood she lived in. He used to hold on to the walls of her house because it reminded him of her. He loved the street she used to walk on.
One day, the ruler at the time came to hear about Majnoon's extreme love for Layla, and so he thought she must be an extremely beautiful woman. He summoned her to his palace. When she arrived, he saw that she was extremely dark and had no attractive features whatsoever. He asked her, "What on earth does Majnoon see in you?" She replied, "O king, you are seeing me through your eyes, so you cannot see the attraction. You have to see me through the eyes of Majnoon..."