A week isn't 8 days, or maybe it's just me who don't use it. But for the 15 days, it's totally true! Plutôt que de dire "On se voit dans 2 semaines?", on dirait plutôt "on se voit dans 15 jours?". But the question was 'why?'. Unfortunately, I don't know maybe it sounds better ahah because it's shorter (to say 'quatorze'/'quinze') but I can't be sure.

Where did you read that? That's so wrong!

A week is 7 days, not 8. Two weeks is mathematicaly 14 days BUT two weeks are also half a month which is 15 days. Instead of saying "on se voit dans quatorze jours", we'd rather say "On se voit dans quinze jours" which is way shorter.

I'm no authority on this (it's the first time I heard it) but I can easily imagine the reason. It is probably 15 is such a nice and familiar number. It is (practically) half of the 30 days in a month, one and a half times of ten, etc. 15 is just more familiar and inviting than 14 which is out there all by itself. Has anyone ever seen the number 14 with a special meaning other than the number of days in two weeks?

It's interesting! In Colombia, instead of "one week later" they also say "8 days later". I'm not sure about the 2 weeks=15 days though.