How did her become the female equivalent of both him and his instead of only being a possessive pronoun like his? Is there a reason?
She likes him and his dog.
He likes her and her dog.
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This is part of a larger phenomenon than just this one pair of words in English. In German, we see the same coincidence with ihr, which is used for both third-person feminine dative pronoun and the third-person feminine possessive determiner. We also see the same forms used in German for the genitive and dative case of feminine singular nouns and articles, as in "der Hilfe" which can be genitive or dative.
It seems to be caused just by an accidental collision of originally distinct forms for feminine singular words. Sound changes eroded the ends of words in ways that caused these two different functions to be expressed by identical forms.
For the possessive pronouns his and her, Wiktionary gives the Proto-Germanic forms as *hes and *hezōz respectively.
The "objective" pronouns him and her are etymologically derived from dative-case forms, which Wiktionary gives as *himmai and *hezōi in Proto-Germanic.
You can see that at the Proto-Germanic stage, the ancestors of possessive and objective her are reconstructed as having distinct forms. Subsequent sound changes would explain why they fell together in modern English and German. Gothic apparently did not undergo sound changes like that.