Reijou Elizabeth no Kareinaru Migawari Seikatsu

One day, Elizabeth Magnus, the daughter of a lowly countryside noble family that recently got into some financial trouble, is suddenly kidnapped in broad daylight by a handsome yet incredibly irritating man she has never seen before. The man turns out to be duke Sylvestor, whose party-crazy sister has run off and is nowhere to be found, is convinced that the woman he found is his little sister, who coincedentally is also named Elizabeth and looks almost identical. Upon realizing his mistake, Sylvestor makes Elizabeth an offer she can’t refuse: she will pose as his missing sister for half a year - or until he is able to find her - and he will provide financial support for her family. So starts Elizabeth Magnus’ life posing as Elizabeth O’Brien, the daughter of a noble.

(Source: Kadokawa, translated)