Recently, an election took place in the Indian state, Bihar. Where JD(U), in alliance with NDA won their elections. Nitish Kumar took his oath as CM within 2 days of the results. Why is it different in the US presidential election?
It's an artifact of historical constraints that have not been (fully) adapted. The timetable for elections is a consequence of a largely agrarian American society at the end of the eighteenth century/beginning of the nineteenth century. Elections were timed to not interfere with the harvest nor with churchgoing (which is why they end up on a Tuesday in November). There was often a need for more remote citizens to make a trip into a nearby town, quite possibly necessitating an overnight stay, in order to vote. Then, sufficient time was necessary to collect all the ballots and convey them to the state election officials who would need to contact the designated electors for them to meet on the Monday after the second Wednesday in December. The votes then needed to be transmitted to Congress for certification by the fourth Wednesday in December with the expectation that if they aren't received by that date, the president of the Senate (or the archivist if the president is unavailable) can contact the relevant state officials to make sure that they deliver the votes by registered mail. The official tally of votes is done by a joint session of Congress on January 6th. Until 1937, the inauguration was held on March 6th to give time for the president-elect to make it to Washington and begin the transition. In 1937, the constitution was amended to truncate the time and bring the inauguration to January 20th.
TLDR: It's an artifact of a time when there were no telephones and all communication and travel happened through horse-powered travel.