Contrary to what Olympian said, while you can't force such positions as White, the French as Black has a close companion against 1.d4, and that is the King's Indian Defense. Here's why:
1) Both apply the "Theory of Pawn Pointing" - When the center is completely blocked by pawns (i.e. 1.e4 e6 2.d4 d5 3.Nc3 Bb4 4.e5 c5 5.a3 Bxc3+ 6.bxc3 Ne7 or 1.d4 Nf6 2.c4 g6 3.Nc3 Bg7 4.e4 d6 5.Nf3 O-O 6.Be2 e5 7.O-O Nc6 8.d5 Ne7 9.Ne1 Nd7 10.Bd3 f5 11.f3 f4 12.Bf2 g5), the theory of pawn pointing applies. Attacks will occur on the flanks, and the direction in which your pawns point dictate which flank you should be attacking. So for the French Winawer (or MacCutcheon or Advance would also apply), that's Kingside for White, Queenside for Black, while in the Classical King's Indian, that's Queenside for White, Kingside for Black.
2) Both feature the same downfall for Black, the Bad Bishop. In the French, it's the LSB, in the King's Indian, it's the DSB.
3) Both are agressive defenses.
I am NOT a firm believer in "sister openings" as they are NOTHING alike. "Sister Openings" meaning same pawn structure for Black, like the QGD and French, Caro-Kann and Slav, or King's Indian and Pirc, they play nothing alike.
A French player should be playing the King's Indian. Same blocked center. Same pawn-pointing themes. Etc.
At first, I was really not a big fan of the French because of the light square bishop being hemmed in. But after giving it a try I feel really comfortable with the opening, win or lose. I've notive that there is very rarely ever a time where I feel I don't know what to do. Well, I was wondering what other openings are similar to the french both for White and as a defense against d4?