One characteristic is the exploration of different scenarios. Some fanfictions might place them in alternate universes where the events of the movie didn't happen as they did in the original. So, they could meet under different circumstances and their love story would unfold in a unique way. Also, there is usually a focus on their emotional connection, like how they communicate their feelings, especially given Elsa's powers and her past of isolation and Hans' complex motives in the original story.
In Elsa and Hans love fanfiction, often there is a lot of character development. Since their initial relationship in the movie was so rocky, the fanfictions usually focus on how they overcome those initial negative impressions. For example, Hans might be written as having a change of heart more thoroughly and Elsa learning to trust again.
It might start with a chance encounter. They see each other in a new light and gradually start to develop feelings. Their relationship develops as they exchange kind words and small gestures of kindness.
One common theme is redemption. Since Hans was initially the villain, a lot of fanfictions focus on him redeeming himself in Elsa's eyes and winning her love. Another theme is the power of love to change people. For example, Elsa, who was so closed off, might open up because of her love for Hans.
Their relationship would likely become more solid. Having a baby together means they have a new shared responsibility. They would have to rely on each other more, like when taking care of the baby at night or making important decisions about the baby's future.
One popular Elsa and Hans love fanfiction is 'A Different Beginning' which explores what could have happened if Hans had truly reformed and he and Elsa developed a relationship in a more positive way. Another is 'Frozen Hearts Reunited' where they meet again after the events of the movie and gradually fall in love through various challenges they face together.
One possible plot could be that Elsa and Hans meet secretly. They start to share their true feelings and dreams away from the prying eyes of the kingdom. As they fall deeper in love, they face the challenge of how to make their relationship acceptable to the people who have misjudged Hans in the past.
The idea of an Elsa and Hans love story is a false one. In the Frozen universe, Elsa is a complex character dealing with her own insecurities and the power she has. Hans is a conniving man who wants to take over Arendelle. He fools Anna into thinking he loves her, but there's no relationship with Elsa. Elsa is more concerned with her family and her place in the world, and any thought of a relationship with Hans is completely out of the question as his true nature is revealed to be quite malicious.
Anna could be the one to save Elsa. After all, she has proven herself brave and determined in the original story. She would stop at nothing to rescue her sister from Hans's clutches.
Anna would be a likely candidate to rescue Elsa. She loves her sister deeply and has shown great courage in the past. She would stop at nothing to find and free Elsa from Hans.
One possible plot in this fanfiction is that Hans kidnaps Elsa not just for power or revenge, but because he is being controlled by some dark magic. He takes Elsa to a remote location where she has to fight not only Hans but also the source of the dark magic to free herself and save Arendelle. Along the way, Elsa might discover new aspects of her powers that she didn't know before.
One key element is the exploration of their relationship. Since in the original story their relationship is quite complex, fanfiction often dives deeper into how they might interact on a more personal and romantic level.