I live in France and I think a more accurate title would be : although almost no one remembers it, syria was actually shortly administered by french authorities a long time ago.
I do not know anyone in France who thinks syria is "french" . If I recall correctly there was only this short lived mandate from the League of Nations after world war one and that's all.
Since The "empire" has almost always been unpopular in france, I doubt a lot of french student took any interest in this very specific and short part of our history. The idea that france used to have some legal responsibilities regarding the defense of christians and minorities in the middle east, is also ignored by the most people.
The relationship with lebanon is a bit more well known but the exact historical context is rather blurry for a majority french people. I asked my young cousin about syria, he thought it used to be an english colony but on the other hand he knew quite a lot about the mosaic of minorities ( christians, alaouites, druzes, Kurds etc ...) in the region, probably because the topic was often brought up in the french media during the numerous Lebanon's "wars".
The thing that help a bit the few who supports an intervention is that we are dealing with actual weapons of mass destruction. I am not sure if americans perceive those weapons the same way a majority of french/europeans might. During the first world war in europe, gas became synonymous with chaos and barbarism ( and there horrific use during the second one reinforced this idea ). This aspect of european history is more likely to affect public opinion regarding this issue than the european mandates in the middle east.