He might be jealous because he secretly loves her. He doesn't want to see her with other people.
Maybe Sasuke has always thought of Sakura as his in some way. When he sees her showing interest in someone else or getting attention from others, it makes him jealous. He watches her jealously as he's not used to sharing her attention and deep down, he wants her to focus only on him.
One possible plot could be that Sasuke sees Sakura getting close to another male character. He starts to feel jealous but doesn't know how to handle it. So he just watches her from afar, his eyes filled with a mix of envy and longing.
In some fanfics, Sakura might avoid Sasuke because she's tired of his cold and distant attitude. She wants to surround herself with people who actually care about her and value her, so she chooses to stay away from Sasuke to find more positive relationships.
Well, it could be due to his dark past. Sasuke has been through a lot and sometimes in fanfics, that pent - up anger and trauma might make him lash out at Sakura. He might see her as an easy target because she's so close to him.
Sakura's ability to forgive and still care for Sasuke despite his past actions could also make him proud. In many fanfictions, she represents unconditional love. This selfless love and her ability to see the good in him, even when he was at his worst, could make Sasuke proud of her as a person.
Well, perhaps in this fanfic, Sakura is a spy on these seduction missions. The jealousy part could come in when her partner or a friend starts to feel left out or envious of the relationships Sakura is building during her missions. For example, Sakura might be getting closer to a powerful figure as part of her mission, and her teammate, who has feelings for her, becomes jealous of the time she spends with that figure.
Well, it could be that in certain fanfics, the writers want to create a more complex and angsty story. Sasuke's character is often seen as a brooding and driven one. So, they make him hate Sakura perhaps because they think it adds an interesting dynamic. For example, if Sakura is constantly trying to get close to him but he hates her, it can lead to a lot of conflict and emotional turmoil in the story.
Another reaction could be that Sakura tries to follow Sasuke. She believes that she can change his mind or that she can be of help to him. She might leave her own responsibilities behind for a while to chase after him, thinking that their bond is strong enough to overcome whatever is making Sasuke leave.
Sakura would probably be initially very angry. She has strong feelings for Sasuke, but kidnapping is a very extreme act. So she would likely fight back, using her medical ninjutsu skills and combat abilities to try and escape.
In the Naruto fandom, such a fanfic would be very out of character for both Naruto and Sasuke. They are comrades and friends with Sakura. However, some dark or AU (Alternate Universe) fanfics might explore extreme and unorthodox ideas. But in the main canon, they would never do this as they have a deep bond with Sakura. Naruto has been friends with her since childhood and Sasuke, despite his aloofness, also values her in his own way.
Maybe Sasuke sees Sakura's growth as a kunoichi. She has become strong and independent, and he admires that. So he wants her to be his.