My kids and their friends say:

Chips !

then, if that was simultaneous again:

Double chips !
…
Triple chips !

and so on until one wins.

When more than two people say the same thing at the same time, one might also say:

Chips personnel !

to win the “contest”.

The loser has to stay mute until his/her first name is pronounced by anyone.

Here is a reference: https://warriors-en-savates.skyrock.com/1309936278-La-Loi-du-Chips.html

A “micro-poll” shows that this expression is known by young people between 11 and 20 years old in both Paris area and Brittany but mostly unknown by older people.

I don’t know if there is an “official” word for this situation.

But, your question reminds me, when i was a child, we used to say “Chips” simultaneously.

So, if our “Chips” was originally a distortion of your “Jinx”… This is very funny.

In my extended family, we say

Les grands esprits se rencontrent

(I’ve never heard of “chips” in the other responses.)