What is the Sharia ruling on changing the wife's name from "Karima" to "Farah" after an animal was slaughtered for her by the name "Karima," and is it obligatory for me to convince my family and her family of the name written in the official documents, which is "Karima"?

There is no blame upon the wife for being called by either of the two names. Multiple names are permissible as long as they do not entail anything unlawful. The Prophet, peace and blessings be upon him, had many names, and five of them became well-known, which he mentioned in the hadith: "I am Muhammad, and I am Ahmad, and I am Al-Mahi (the obliterator) by whom Allah obliterates disbelief, and I am Al-Hashir (the gatherer) by whom people will be gathered at my heels, and I am Al-'Aqib (the last)." Therefore, there is no blame upon either family for calling the wife by either of the two names.