The connecting sense is the claw of an animal, which is narrow and tapers to a point in a way human nails do not. [The PIE root from which we get the word *nail*](https://www.etymonline.com/word/nail) gives rise to many words also meaning *claw*.
The etymology here is actually pretty weird: a common PIE root split into two words in Old English, which then re-combined later in the history of English. The same seems to have happened in other Germanic languages.
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