First of all, may I remind you of John 1 again: nothing was existent before Christ, not even the Bible or any of its versions and translations, including the Masoretic text or the Septuagint.
That said:
The oldest fragments of the Septuaginta we have are from the 2th century BC.
The oldest fragment that we have of the Gospel of John is Papyrus P52, dated at around 100-150 AD. So as for your question what makes such manuscripts a ‘Septuagint’: Seeing that the pericope of Jesus and the adulteress (John 8:1-11) doesn’t show up in any its earliest textual witnesses, you may as well ask what makes the Gospel of John the Gospel of John.
The oldest still existing codices (originally) containing the entire Bible with (pretty much) all the Books of our Canon(s) in them, are also ‘only’ from the 4th and fifth century. They are the very
Codex Sinaiticus,
Codex Vaticanus, and
Codex Alexandrinus that I already mentioned as containing the Septuagint. So as a ‘bible-believing’ Christian you may want to be a bit more careful with your criteria, otherwise you risk cutting the bow you are sitting on.
However, should you ever come across skeptics/atheists who tease you about the Bible’s oldest copies being rather ‘young’, tell them that the oldest manuscript we have for the writings of Tacitus is from the 9th century, and yet they’d never doubt the reliability of such secular textual transmission
http://ho-logos.blogspot.de/2009/02/canon-textual-criticism-and-more-with.html
Not that I’m a massive nerd in the field of textual criticism myself, Stranger, in fact if anything I’m the proverbial one-eyed man here. But I really wonder why it is that so many Christians who pride themselves to be the most ‘bible-believing’ are apparently utterly uncurious concerning the question how its texts got to us in the first place.