First, a brief lesson in historical linguistics. Language families arose at some point in the past from what linguists call proto-languages, which is a term that refers to a language that is reconstructed rather than one that is attested. There is debate over whether language families all derive back to a single proto-language or if several proto-languages developed independently. In either case, as the group forming the speakers of a proto-language grows over time, it eventually splits into two or more separate groups whose language started out the same and then over time became mutually unintelligible. Those two “child” languages are related in the family tree of languages. The child languages can then grow and split as well. In the history of English, there have been several splits, but two are significant: first, the proto-Indo-European language split into various languages forming the subfamilies of Indo-European, such as the Germanic languages (proto-Germanic) and the Romance languages (Latin). The Germanic languages also split, eventually resulting in English being a separate language from the other Germanic languages, such as German, Swedish, and Dutch.
A word becomes part of a language in two ways: (1) it inherits them from ancestor languages; (2) they borrow them from contemporary foreign languages. For example, the English word brother derives in a direct lineage back through proto-Germanic to the proto-Indo-European root *bʰréh₂tēr. This is an inherited word. The other way a word enters a language is to be borrowed from a contemporary foreign language. This is like the English word croissant which was borrowed from the French word croissant. That is, when the word croissant became a part of English, it was copied from people who were speaking a foreign language—French. Even though French and English are related—both being Indo-European languages—the word croissant was borrowed from French because the word entered English after French speakers and English speakers split apart long enough for their languages to be not mutually intelligible.
Now, French also has a word for brother that derives from the proto-Indo-European root *bʰréh₂tēr: that French word is frère. This is where the word akin comes in. The English word brother is akin to the French word frère because both languages inherited those words from their ancestor languages, and they both ultimately derive from the same root in their closest common ancestor language, proto-Indo-European.
So, when a dictionary etymology says “word A in language L1 is akin to word B in language L2”, that means that neither word A nor word B were borrowed into their respective language and both word A and word B derive from the same root word in L1 and L2’s common ancestor language. That is, when Merriam-Webster say about salt that it is “akin to Old High German salz salt, Lithuanian saldus sweet, Latin sal salt, Greek hals salt, sea” they mean that the English word salt and those other words from related languages all come from the same root in their common ancestor language, proto-Indo-European.
A couple of other terms you might find:
- derivative: a derivative is a word that is derived from a root word. Both brother and frère are derivates of *bʰréh₂tēr.
- reflex: a reflex is a sound in a descendent language that corresponds to a sound in the ancestor language. For example, the proto-Indo-European sound /bʰ/ has the reflex sound /b/ in English and /f/ in French, which is why the English derivative is brother and the French derivative is frère. Similarly, the proto-Indo-European root *bʰudʰ-no- has derivative bottom in English and fond in French, and *bʰerǵʰ- meaning “fort” has derivatives burg and borough in English and fort in French. (The English word fort was borrowed later from French).