Before I get to the essence of your question—whether or not sie and sein are etymologically related—let me get out of the way that no translation of the Bible has any effect on that matter.
The oldest accepted translation of the Bible into German was made by Martin Luther in the 16th century. By this time, the language spoken in the area now known as Germany is already known to present-day historic linguists as Neuhochdeutsch or modern high German. Before that, in the Middle Ages, Mittelhochdeutsch or medieval high German was spoken and even prior to that Althochdeutsch (old high German) was the collection of dialects at the beginning of the Holy Roman Empire. The German language thus already had centuries of evolution meaning pronoun words were known and fixed by the time Luther translated the bible. Furthermore, the original text that Luther translated is probably at least two millenia older than him and was originally written in Hebrew. Luther might have punned while translating the Bible but that cannot be used to construct etymological relations.
To answer the actual question, a peek into the etymological dictionary by Wolfgang Pfeifer, which is made available by DWDS.de is often helpful. However, the sentences are often hard to understand, even for natives. This dictionary traces sie, the singular pronoun, back to an old pronoun stem s-. This stem is noted to have been used also as an alternative form of the demonstrative pronouns (now der/die/das) but these ‘were given up early and replaced (in old high German prior to tradition) adapted to the other cases by analogy’ (source). A lot of the additional text concerns how the formal forms changed over time.
Sein, the pronoun meaning his, is traced back to the root *se ("him-/her-/itself"), whose locative *sei is adjectivised by the suffix -no. While si and *se may look similar, the text does not note this relationship nor any reference to the demonstrative pronouns.
So without diving all too deeply into the labyrinth of etymology, it already seems rather clear that the two derive from different roots that probably trace back to different Proto-Indo-European words. Thus, there is no relation.