In the 1850's and early 1860's, dark floral patterns had come into style in the United States. Carpets went from wall to wall, and draped from the ceiling to the floor. Dark colors were en vogue during those years, as well as bold patterns for the wallpapers. Mary Todd had remodeled the Lincoln family house in Springfield after these fashions, and when they moved to the White House, she decided to remodel it as well.

According to David Herbert Donald, the White House was in a "bad shape. The furniture was broken down, the wallpaper peeling, the carpeting worn, and the draperies torn. The eleven basement rooms were filthy and rat-infested. The whole place had the air of a rundown, unsuccessful third-rate hotel." (This means that when Mary talked with Stevens of the bad conditions of the White House when she arrived, the film was being historically accurate). Congress decided to appropriate 20,000 USD to remodel the executive mansion, and Mary spent not just the yearly allowance, but the allowance of Lincoln's entire 4 years, buying furniture, fabrics and wallpaper. Most of them were dark colors, with bold patterns. For example, she bought crimson carpet for the "Red Room". Other rooms had lighter colors, such as an imported velvet carpet of a pale green color, which one visitors described as looking like the sea.

Mary had never been a conscious buyer, and she had a bad habit of hiding her purchases from her husband. When she discovered that she had so greatly overspent, she tried to sell second hand furniture, pad bills for domestic purchases, or act as the White House stewart, receiving the salary. But, of course, this could not cover her expenditures, and at the end Congress had to appropriate money to cover them. Honest Abe was furious. He said that he could not ask Congress for funds “for flub dubs for that damned old house!” because “It would stink in the land to have it said that an appropriation of $20,000 for furnishing the house had been overrun by the President when the poor freezing soldiers could not have blankets.”

But I digress. Most of the rooms were indeed dark. The producers, as far as I can tell, did a magnificent job researching how the White House should be decorated. Take into account, however, that back in those times they did not have electricity like we do, and the producers emulated that for the most part, making the rooms look even darker. Also, it's possible that they consciously made it appear dark to emphasize how Lincoln is going through a dark hour. Note than when the Amendment passes, Lincoln opens a big door, allowing light to enter into the room.

As for why this style was popular, it seems that using dark colors was a reaction against the excessively shine and colorful clothes and decoration of the 18th century. Dark colors projected an image of sobriety, decency, and restraint, that became popular during those times in Europe and the US. You can see this in how men's suits were often black, while 18th century aristocrats used jackets and pants of all colors. I'm admittedly not an expert when it comes to fashion, so I'm sure that someone else can give a better answer than me in this regard. Nonetheless, as far as I can tell, dark colors were indeed popular in the 1860's, and Mary remodeled the White House using dark colors, making the film historically accurate.

Sources: David Herbert Donald's Lincoln biography talks a little about the remodeling and decor of the White House, and the fashions of the age.