French defence like vs 1.d4
Why?
What's the purpose of wanting this?

So a resilient defence with asymmetrical chances for black ?
I would say the Nimzo/Queens Indian complex comes into consideration. Then there are the "annoying sidelines" such as the London eg 1d4 Nf6 2Nf3 d5 (not compulsory) 3Bf4 you have to prepare for.

The Dutch or the Tarrasch Defence of the QGD are good choices for you if you play the French against 1.e4. The pawn structure in those two openings have greater similarities with the French than does the Nimzo and Queens Indian (both excellent openings). The advantage of the Dutch and the Tarrasch are that they can often be used against 1.c4 and 1.Nf3 as well.
Why?
What's the purpose of wanting this?
Ok, what structures do you play in your French games?
Off the top of my head, French-Caro comes to mind
For example:
1. e4 e6 2. d4 d5 3. Nc3 dxe4 4. Nxe4
1. e4 c6 2. d4 d5 3. Nc3 dxe4 4. Nxe4
I think there's a similar King's Indian formation that can appear in the French
. But yes, Caro-Slav structures are similar.
What about your white repertoire structures?

Yes you can play 1...e6 against 1.d4 too. If you do not mind that white make a French of it as you play French on 1.e4 anyway.
After 1.d4 e6 white can make it a French but that s no problem for you but mostly he will play 2.c4 now.
And then of course you could play 2...Nf6 and go for a Nimzo-Indian or Queen Indian or also a Queens Gambit declined or even a Benoni. But you can also go for 2...f5!? and play the Dutch with the avantage that a Stauton Gambit is not possible anymore. Thus 1.d4 e6 2.c4 f5 is interesting you can play Nf6 later. Also on 1,c4 you can play 1...e6 too.

If you play the French maybe as white you could play something like this. 1.e3 Nf6 2.Nf3 g6 3.d4 Ng7 4.Be2 0-0 6.0-0 d6 6.c4 in a bit French style. Van Kruis opening.

RubenHogenHout: Totally agree with you. Well worth replying 1...e6 to 1.d4, and then transposing to the Dutch against 2.c4 or 2.Nf3. 1...e6 also has the advantage of avoiding one of the more annoying sidelines against the Dutch, 2.Nc3. Then 2...d5 leaves White with a version of the Richter-Veresov where s/he has to play Nf3 to get Bg5 in.