There is an art to spotting elements in paintings of bygone centuries and reading meanings into them: it's called iconography, and any young art historian spends a lot of practice time looking for those elements and researching the meanings thereof.
Dogs are easy, for the most part. When shown near a woman or a married couple, the dog is shorthand for: this woman is a faithful wife; this couple will be true to each other. In Catholic artworks depicting St. Dominic, founder of the Domincan order, it can be a play on the Latin domini canis, "dog of the Lord," and those familiar with the Dominicans' role in converting various populaces can reflect on how apt a metaphor that may be. (Here's a good illustration and explanation of that.)
Dogs also symbolize courage and vigilance, just as we prize them for those qualities when they watch our houses.
Do you want a look at the sheer dazzling array of intellectual work ahead for you as a budding (dog) iconographer? Check out this article.