Depending on what database you look at, all three moves (5.c3, 5.Nc3, 5.Bd2) are playing in this position. Nc3 and c3 are the most common, but Bd2 is played sometimes.
A lot of the play in the French Exchange is centered around the central squares and the open e-file. Quite often, White will play c4 and Nc3 and try to deploy their bishops at d3 and f5 or g4.
The problem with playing 5.Bd2 in my opinion is that the bishop is much stronger on a square affecting the center rather than trying to counter Black's bishop. Additionally, if Black plays 5...Bxd2, White can recapture with the Knight, which would put the knight slightly out of position (as it's better on c3 than d2), or the Queen, which would put it slightly out of postion (it's not really affecting anything on d2). If Black does not capture and plays like 5...Bd6 (it's best square), then White needs to reposition the bishop, which is doing nothing on d2.
Playing 5.c3 is fine. It moves Black's bishop, has central solidity, but creates a gameplan where white will need to do more maneuvering to get its pieces (i.e. the b1 Knight) into good positions. Ultimately, the Exchange French tends towards a draw and passive play, and, while 5.c3 seems active, it creates a passive setup for White.
I think @cyclops is right in the value of 5.Nc3. It keeps up the pressure on the center and develops a key piece to its most influential square. Yes, Black can take it and double White's pawns, but a) Black's dark-squared bishop is a really valuable piece in this opening, b) it still allows white to play c4 comfortably to hit the center. It may look messy, but it's better overall.
The French Exchange is usually fairly slow and equal. Bd2 may put White a little out of position, but it does depend on what kind of game White wants to play, i.e. if White wants to be active, passive but solid, or simply counteract anything Black does.