In the 21st century, English speakers usually use the word bane the way Anthony did in "Bridgerton" as part of a phrase beginning "the bane of...," and it's often used ironically or to exaggerate how awful something is.
The Urban Dictionary points out that it's "something that is so disagreeable with your spirit that it feels like its existence might negate yours."
The laser pointer is almost certainly the bane of your cat's existence. Some people love avocado, while the hidden presence of guacamole in a burrito is the bane of someone else's lunch. A full email inbox is almost certainly the bane of one's workday. Or it could be used to describe a legitimate problem, like the fact that icy sidewalks are the bane of dogwalkers in the winter.
In "Bridgerton," Anthony was telling Kate that she was a royal pain (so to speak), but he couldn't keep himself from obsessing about her.
It's a real mixed bag, but it's the perfect thing for Lord Bridgerton to say to Miss Sharma as their relationship moves from enemies to lovers. However, calling someone the bane of your existence is not often the most romantic move.