The dog is actually a classic symbol of loyalty, as it is very well known that dogs are extremely loyal to their owners, however this is not the sole explanation for it.

The Museo del Prado was created from the Royal Collection of the Kings of Spain, who were big mecenae, patrons of the arts. Both the Habsburgs and the Bourbons employed different Court painters to paint pictures of the family, and also of individuals of the royal family as individual portraits. Velázquez, for example, was not only the Court painter, but also a good friend of King Philip IV, which resulted in a number of paintings of a not so royal style, but more of a natural attitude.

King Philip IV was an avid hunter, as many members of his family before and after him. Hunting was an activity linked to royalty and nobility since time immemorial, as can be attested by other works of art like the sepulchre of Fernán de Andrade the Good, in the church of Saint Francis in Betanzos (Galicia), that rests on top of two massive figures, one being a female bear, and the other one a female boar.

Charles V was also a big fan of hunting, and in one of his portraits you can see him accompanied by one of his dogs. Philip II, son of Charles V, enjoyed hunting too. In fact, one of the reasons for choosing Madrid as the place for the Court was that it is close to the royal mount of El Pardo, abundant in game, so the king could go hunting without being far away from the Court. There is even a painting of Giovanni Andrea Doria, grand-nephew of Andrea Doria, with the dog Hercules, that had been a gift from Philip II.

Philip IV, the cardinal-Infante Ferdinand of Austria, King Charles III , prince Baltasar Carlos, have all been painted in hunting attire at different points, and of course accompanied by their trusted hunting dogs, a very necessary tool for hunting.

So, the dogs that you see in paintings were actual dogs, loyal companions to the Kings, that not only represented loyalty, but also royalty and wealth, for maintaining a hunting dog, or a pack of hunting dogs, was expensive. Of course you may have also seen mythological scenes in there, like the death of Acteon, and that requires, by its own nature, the presence of several hunting dogs.

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Edit: Added links to the images of the paintings I mentioned