Ben is a Turkic word, not a loanword. It's descended from Proto-Turkic *ben.
In just about all of the Turkic languages, it is either ben or men or something similar with a vowel change (bän, min etc.)
I can't find any sources, but I'm assuming the Persian man is simply derived from the oblique forms of the first person singular in Proto-European languages, just like the English word "me" is. The similarity is surely a coincidence, possibly partially encouraged by a Sprachbund effect.
By the way, the English words they, them, their and theirs are loans from Old Norse. Borrowing pronouns is rare, but it does happen sometimes - just not in Turkish.