Yes and no (I have played both as black )
Yes the play is similar as in the pieces go to the same squares and the same center squars are important. No in the sense that the e-file is open leaving less for white to play for (no minority attack, no real chance of a kingside attack). So (my subjective opinion) against the CK it is a winning try, but against the French it promises very little (the lines pointed out by Dom and some others). In fact I havent spent much time on this variation of the French and based my line solely on the French book by Uhlmann (1 game) and so far havent run into trouble with it. I face it at least once a year, because one clubmate plays this line with white, so I always play the french against him. So far 4-0 score in my favour.

No problem with opening a new thread about c4 in the Exchange..
About the statistics: if you look at chessbase online database, and search for Miezis's games with c4 and opponents rated above 2500, you'll see no glory for Miezis ...
Miezis played some weird moves in the games: 4...Nf6 5.Nf3 Bb4+ 6.Bd2 (Watson marked it as dubious, exchanging the bishop in an isolated pawn position) or 4...Nf6 5.Nf3 Be7 6.Be2 (the only advantage is to let Black choose the system, and maybe a good one is strong point system like in Miezis-Short,Leon 2001...but dxc4 is still playable and transpose in some "main lines" with Bxc4: 5.Nf3 Be7 6.Be2 oo 7.oo Nc6 8.Nc3 Be6!? Miezis-Short,Leon 2001 and light-square bishop was inactive for White)

Be7 maybe viewed as a "safe" line comparing it with Bb4, but the move and the system can be played like in:
4...Nf6 (if 4...Bb4 White can play Nge2 instead of Ngf3 newt moves) 5.Nf3 (5.Nc3 Be7) Bb4+!? 6.Bd2 (Miezis's move ; 6.Nc3 is normal: 6...oo 7.Be2 dxc4 8.Bxc4 Re8+ 9.Be3 Be6!?) Bxcd2+ 7.Nbxd2 oo 8.Be2 dxc4 9.Nxc4 Nbd7 10.oo Nb6 11.Ne3 Be6 12.Ne5 Qd6 Ellenbroek-Stellwagen,Wijk aan Zee 2001 (Watson)

I am revisiting my handling, as White, of the French exchange. I will admit that I sometimes aim for a draw by choosing this line, but the game does not always end up being drawish at all, which is fine with me.

Anyway, my query is about the status of 4.c4. I have indeed noted that if 4.Nf3 and 4.Bd3 are played about one game every three, 4.c4 is played about one game every six and by as highly rated players as 4.Nf3 and by higher rated players than 4.Bd3. Last but not least, it has a success rate of 52% against 46% for both 4.Nf3 and 4.Bd3 (according to ChessBase's 2006 Opening Encyclopedia). The main proponent of 4.c4 is GM Normunds Miezis (2529 FIDE ELO) who wins two thirds of his 4.c4 games, which is not bad at all.

So far, I have failed to find a proper assessment of this line and I wonder if it is worth investigating further.

Any thoughts or, better yet, hard facts out there ?

Many thanks !