I agree. My taste buds LOVE French Vanilla and I agree it's a more intense Vanilla taste. I use Davinci French Vanilla coffe syrup in my all butter buttercream recipe and it's divine.
On the other hand, my mother (who hates eggs) says that French Vanilla (as in ice cream) is too "eggy" for her.
Guess it depends on who is tasting it! 
P.S. Duncan Hines makes a French Vanilla cake mix. Is your customer thinking along those lines?
Out of curiosity for an "official" difference, I googled and found this from wikipedia:
The term French vanilla is often used to designate preparations that have a strong vanilla aroma, and contain vanilla grains. The name originates from the French style of making ice cream custard base with vanilla pods, cream, and egg yolks. Inclusion of vanilla varietals from any of the former or current French dependencies noted for their exports may in fact be a part of the flavoring, though it may often be coincidental. Alternatively, French vanilla is taken to refer to a vanilla-custard flavor.[19] Syrup labeled as French vanilla may include custard, caramel or butterscotch flavors in addition to vanilla.
Well there ya go! My mom is right, apparently it IS eggy! 
I knew the reason I was loving that coffee syrup is because it gives a more complex vanilla taste with a hint of caramel!
I know in ice cream, french vanilla is custard based ice cream where vanilla is not however, not sure how that would translate to a cake. I would probably go with a richer cake recipe (maybe one with yolks vs. whole egg) and, a stronger vanilla flavor.
I bought my wine sized glass bottle for less than $10 at TJ Maxx. Oh I how I wish I had bought 2 while they there! It's been ages and they haven't gotten them again.
Here is an Amazon link to what the label looks like:
http://www.amazon.com/dp/B000UI6U10/?tag=cakecentral-20
Not sure if it has to do with your request at all, but Toba Garrett has a French vanilla buttercream. And it is regular vanilla added to it. It's what Chasey said before, it also refers to a preparation, made with a custard.
http://cakecentral.com/cake-decorating-ftopict-19577-.html
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