I'm an English speaker learning Russian and from the little bit of French that I know the negation construct "ne pas" feels quite similar to "не раз".
Is there in fact a historical connection between these two negation constructs?
The constructs are not linked etymologically.
The French pas comes from the Latin word for "step" and the constructs like "ne … pas" originally literally meant "don't (move, walk etc.) a single step", and were only used with the actual verbs of motion. Pas has later grammaticalized (become a function word rather than a content word) and is now used with all verbs, not only verbs of motion.
The Russian раз originally meant "strike" and is cognate to разить ("to strike", "to hit"). За один раз originally meant "in one strike", then the metaphor had worn out and the word's original meaning had been forgotten. This word is still a content word (meaning "time", "repetition") but has also acquired a grammatical role (раз ты такой большой, иди работать // "Now that your are all grown up, go find a job").
Those constructs are indeed similar in the sense that they both started as metaphors ("a single step" in French, "a single strike" in Russian) which later lost their metaphorical meanings and became integral parts of their respective languages. However, there is no historical or etymological connection between them, those are parallel independent developments.
I don't think there's any connection between these two phrases. What attracts attention is a mere coincidence in written forms,the visual image, so to say. Though going deeper into the matter, we can admit that the meanings are different. If the French phrase is translated "not", the Russian one is " not once,more than once". But in the whole, we can say that since French was more popular with the nobility than any other language in the 19th century, and because of the war of 1812, Russian got borrowings from French.