There is one thing about the French that may be an indicator for what s/he may play against 1.d4.

The French is an active, counter-attacking game that relies on an intricate interplay between pawns structure and piece play.  I don't know about others, but I look for very active play as Black without sacrificing a certain fundamental respect for the advantage of the first move.   

This means that I look for an opening that has a track record for results.  I don't mind defending for a while if the resulting positions give me chances to win!  For me, the Slav fits this description perfectly, but I can see other Francophiles picking up completely different openings that also fulfill the basic requirements of being active w/o going over the top.

Its hard to pin these French defence players down, they can play virtually anything with black and usually do.

So far I haven't noticed any particular pattern or profile that fits French defence players, for instance Radjabov plays French and Kings Indian, while Korchnoi plays French but wouldn't dream of playing the KID.

There you have it, clear as mud.

Toppylov

I play the slav. I've tried the Chebanenko once, but recently
white has been having good results with the c5 advance, so 
now I dare not play it again. Instead I go for 4.-,dc4 hoping
for something like the main variation with a black bishop on
b4 (the Winawer connection again) and a generally solid game.

Yeah, in my head I've always grouped the light-square defences together, and the dark-square defences.  So the Nimzo/Bogo/QID I compared to the French and IZ & Stonewall Dutch and some of the QGD structures.  Then on the dark side I grouped the KID and Benoni and Dragon and Pirc and Leningrad Dutch, all of which can end up looking like each other.  It almost seems like a waste of energy to learn a dark-square defence against d4 and a light-square against e4.   

Obviously I'm wrong and over-simplifying.  I've a feeling Watson and Pshakis know more about these things than I do!

I've always thought there to be a kinship between Frenchand Benoni. Or does it only look that way because of the most prominent authors on these defenses, Watson and Psakhis ?

I play the Dutch against 1. d4, through the 1...e6 move order.  The Stonewall is my favorite variation of the Dutch.  I like locked centers.  In the past I've also played the Benoni.   

And though you haven't asked, against 1. c4, I play 1...e5.  Sometimes I play this with an early ...f5, when it resembles the IZ Dutch.  Sometimes I'll play the Reversed Dragon.  I like the position as White or Black.

regarding pawn structure and the nature of position, I think Slav, especially the a6 variation (Chebanenko). Like the French expert GM Sergey Volkov used it succesfully.

I've noticed that the Dutch and the King's Indian are typical favorite defenses against 1.d4 among French players.  (Makes sense, right?  The counterattacking spirit associated with black's asymmetrically confrontation to white's central occupation is a strong similarity.  For example, the King's Indian often involves counterattacking within a central pawn chain structure, as does the French.)

In my experience, those who play the Winawer also like the Nimzo/Queen's Indian or a Modern Benoni instead of the QI after 3.Nf3. 

Those who prefer 3...dxe4 or 3...Nf6 are more likely to go for a Queen's Gambit or the King's Indian.