“Man’s best friend” is a colloquialism, so it sort of depends on what you mean and how seriously you take it. In a very real sense, dogs have been man’s “best friend” since their domestication, about 30,000 years ago. There are arguments that hunting with dogs increased efficiency to give humans surplus time for the flowering of culture. There is evidence that human behavior and physiology co-evolved with dogs, with similar changes to key enzymes in digestion and brain neurochemistry appearing more or less simultaneously in both human and canid lineages.

So in a very real sense, dogs have been “man’s best friend” for a very long time. They were the first animals domesticated; they had a “friend” role rather than merely a mobile meat source, in that they helped with hunting and guarding human encampments, and were socialized as “friends”.

But if you mean as a modern marketing phrase, you’re right that its popularity only goes back to the 19th century. In the early days dogs were bred for a particular need – hunting, tracking, guarding, retrieving, herding. These were working dogs. Of course, there’s nothing to say that a working dog can’t also be a friend, but if you look at breeds that were created solely or primarily for companionship, then most of them go back no earlier than the 19th century. Modern dogs are very mixed in their genetics, because crossing them with other breeds happens all the time, either deliberately or accidentally. But the 19th century is recent enough that very good historical records exist, and you can also do genetic analyses to see how breeds relate to and are derived from each other to create trees of phylogenetic relationships. And you can see that “ornamental” or “companionship” breeds are quite recent.