The various epithets, patronyms and descriptive names are surnames. That's the way all our surnames originated.
The difference between Hobbits and the rest is that they use family surnames.
Family names are important for inheritance. The Hobbits had a system of personal land ownership that passed from generation to generation through the family. Being able to prove one's kinship could become very important if a land-owner died without a heir.
Other peoples in Middle Earth used some sort of feudal system, where all land belonged to the Crown, or a system where land was held in common. In those circumstances, kinship names weren't so necessary.
Another reason why Hobbits needed surnames was to be able to identify themselves individually to use the postal service. They were the only society to have such a thing.
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