Certainly, Lucile’s longevity isn’t a product of Jeanne’s longevity, but it just about possible that both instances of extreme longevity where at the least influenced by the same genetic factors, though this is by no means a guarantee, due to the distance of their relationship.

We are not just talking about the two oldest women in France, but about the two of the four (validated) oldest in world history and actually outliers. Jeanne reached the age of 122 when no one else has yet reached 120 and Lucile was less than 1 month away from her 119th birthday when the 5th oldest person "just" reached 117 years, 260 days. it is really a great coincidence that they were born in such close areas.

Despite the proximity of the birthplace means as you say that individuals with grandparents from roughly the same region, it's more likely to have a common ancestor than not, it is an additional important element to know for sure that they are indeed distantly but traceably connected. So what are the chances of 2 of the 4 world history’s outliers to be related even as distant cousins? Average shared DNA might be a very small percentage but inheritance is random and Calment-Randon could theoretically be more connected than the average or having both inherited a crucial common genetic block.

After all, given their outlier status we already know that they had won the genetic lottery in some way-if they did it the same way it is very difficult to tell based just in genealogy, but i think their distant relation has some possible interest beyond being merely a fun fact.

And as I explained, since they hailed from the same region, and their recent ancestors on all sides also lived there, it's basically inevitable that they would have common ancestors only a few hundred years back in time.

With that said, I'd like to emphasise that the French genealogists who discovered how Calment and Randon are related made an admirable work. But we should look at this as fun trivia rather than assuming this explains their extreme longevity – it doesn't. 

@Tomi : I totally agree with you, but that is really improbable/unlikely that the 2 oldest french EVER are distant cousin !

You're missed, dear Sister André. Rest in peace.