Source and decoding
תנינא: המשכיר בית לחבירו, אם עד שלא מסר לו מפתחות חל ארבעה עשר — על המשכיר לבדוק, ואם משמסר לו מפתחות חל ארבעה עשר — על השוכר לבדוק
We learned: One who rents a house to his fellow, if the fourteenth arrived before he handed over the keys, the landlord must search; if after he handed over the keys, the tenant must search.
Rav Nachman bar Yitzchak was asked who must search and answers from a baraita. The criterion he cites is the handing over of the keys and not mere ownership. The comparison to mezuzah is set aside because mezuzah is the obligation of the resident. So what counts here is who actually holds the house on the fourteenth.