An interesting question, a linguist like Tolkien probably had his own reasoning behind who and why used or did not use last names in Middle-Earth, though I can as well imagine the initial idea being just "what sounded good".
Many people had second names, but only in the sense in which the original surnames for example in many European cultural spaces came to be - just an earned "nickname", like for example Felagund. But still these were individual and never were inherited down in the family - that seemed to occur in all nations only among the Hobbits.
Perhaps some HoME loremasters could say more (I am sure there's been a particular essay about Elves, though I don't think they had last names, even though they could have several names).
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"Should the story say 'he ate bread,' the dramatic producer can only show 'a piece of bread' according to his taste or fancy, but the hearer of the story will think of bread in general and picture it in some form of his own." -On Fairy-Stories
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