English daughter names a female child in relation to her parents, and a little more loosely a female descendant. That kinship sense has been stable for as long as English has written records. What has grown around it is a habit of treating other things as offspring of a source: a woman of a nation or a church, a language split from an earlier one, a cell produced by division, an isotope left after decay.
Native, not borrowed
The origin of the word daughter is English inheritance, not a loan. Old English already had dohtor, an r-stem noun sitting in the same morphological set as fæder, mōdor, and brōþor. Middle English kept the word as doughter, doghter, and a scatter of other spellings while the old velar fricative in the middle of the form faded from speech. The spelling daughter, with augh, settled in southern England in the sixteenth century. The silent gh is a fossil of a consonant that German still pronounces in Tochter.
The immediate ancestor is Proto-Germanic duhtēr, the source also of Old Saxon dohtar, Old Norse dóttir, Dutch dochter, German Tochter, and Gothic dauhtar. English did not pick the noun up from those neighbors. They are siblings of the same Germanic word.
An Indo-European kinship label
Behind the Germanic form stands Proto-Indo-European dʰugh₂tḗr, one of the old r-stem kinship terms that also include the ancestors of father, mother, and brother. The reconstruction is not a single tidy string. Comparative evidence supports a medial laryngeal, the h₂, still visible in Indo-Iranian and Greek. Other branches look as if that laryngeal had already dropped in some case-forms, after which the reduced stems were leveled through the paradigm. Specialists treat that as a known phonological wrinkle, not as two unrelated words.
Cognates run well beyond Germanic. Ancient Greek had thugátēr; modern Greek still uses thygatéra beside the everyday kóri. Sanskrit duhitṛ, Avestan dugədar-, Armenian dustr, Lithuanian duktė, and Old Church Slavonic dъšti (later Russian dočʹ) all point to the same ancestor. Further out sit Tocharian B tkācer, Persian doxtar, and Anatolian forms in Luwian and Lycian. The word was ordinary Indo-European vocabulary for a female child.
Latin is the famous gap. Classical Latin used filia, the feminine of filius (son), built on a different root, and the Romance languages followed: French fille, Spanish hija, Italian figlia. Insular Celtic went another way as well, with Irish iníon and Welsh merch. Older handbooks sometimes say the inherited Indo-European word was simply lost in Celtic and Latin. That is too neat. Gaulish preserves duxtir, and Oscan, Latin’s Italic cousin, has futír. Latin itself replaced the old term; Celtic as a whole did not lose it everywhere.
The milking story, and why it is shaky
Because several kinship labels share a *-tēr ending, it is tempting to read daughter as an agent noun, the one who milks or the one who produces, from a root *dʰewgʰ-. The idea still appears as a cautious perhaps in some reconstructions, helped by a Sanskrit verb that looks similar and can mean to milk. Many Indo-Europeanists reject the link. The milking sense of that root is not securely Proto-Indo-European, and the consonants do not match once the laryngeal in dʰugh₂tḗr is taken seriously. The milking daughter is a possible scholarly guess, not established history. The word is old; its inner anatomy is not agreed.
Senses that grew around the core
The parent-and-child sense never needed replacing. By late Old English the noun could already name a woman in an analogous relation: a daughter of a church, a country, a culture. From about 1200 it was available for anything treated as feminine and derived from a source. That figurative habit is what later technical English still uses. A daughter language is a language descended from an earlier one. A daughter cell is produced by division. In physics a daughter nuclide is what remains after radioactive decay. None of these is a new etymology. They are the kinship metaphor stretched until it names a product.
The compound daughter-in-law is later still, from the late fourteenth century, when it began to crowd out older snoru and the transparent Middle English sonnes wif. That is a chapter in the history of in-law, not a change inside daughter itself.
What the spelling hides
A speaker of modern English who meets German Tochter, Dutch dochter, or Swedish dotter can hear the family resemblance at once. The English spelling, with its silent gh and its augh, hides the same relationship. The useful fact to keep is small: daughter is not a late polite replacement for some lost Germanic word. It is the inherited Indo-European kinship term, still doing the job it had in Old English, while Latin speakers had already switched to a word built on son.