My research shows that this claim is false, dogs see more than just black and white. Dogs don't have the same visual prowess that humans do, but their vision does allow them to see color. Dogs have fewer types of color-detecting cone cells in their eyes, which affects how many colors they can distinguish. This results in dogs perceiving a reduced color spectrum, but certainly more than just black and white.
rspb.royalsocietypublishing.org 2. This source outlined an experiment by a team of Russian researchers who found dogs were much more likely to recognize a piece of paper by its color than its brightness level. The researchers printed out 4 different colors of paper(2 colors, 4 shades), and adjusted the lighting. The dogs quickly recognized change in the colors but not the lighting, suggesting color recognition ability.
https://www.smithsonianmag.com/science-nature/new-study-shows-that-dogs-use-color-vision-after-all-13168563/ This secondary source, part of the Smithsonian magazine written in 2013, discusses dogs having two types of color receptors(cones), and humans having three. The source compares dog vision to humans with red-green colorblindness. This source was based on the experiment of the Russian scientists outlined above.
The Smithsonian magazine has no reason to be biased regarding this issue, if they had any, maybe they'd be biased towards dogs being able to see color, because at the time that went in contrast with popular belief.
Behavioral studies of the past show dogs having trouble differentiating between colors, and it's possible these studies are your only source of information.
Evidence from the Russian experiment undermines this claim, specifically that dogs can see along a blue-yellow spectrum. This study, among others, have confirmed that dogs can reliably distinguish blue and yellow objects from gray ones.