Is there a difference in loyalty? I guess cats are generally more independent and subtle in their attachment than dogs. But as far as actual loyalty and devotion, perhaps I’ve simply met different cats than you, but once they devote themselves to someone, I’ve seen cats be as intensely loyal as dogs (and science supports this). When I said Dog and Cat are good alternatives for Badger and Snake respectively, I wasn’t talking about just loyalty, but other behaviors. My apologies for the confusion; I never fully explained my thoughts.
The first part of it has to do with how dogs live in packs, while cats do not. In a sense, dogs are naturally community minded. So, like how the well-being of the community comes before personal wants for Badgers, most dogs can learn to take commands and do what their humans expect from them even if they'd rather jump on the dinner table and eat everything. Cats, however, will do what pleases them before they’ll listen to their human, like how Snakes will prioritize their personal wants. Oftentimes those wants are to take care of their people, just as you can have incredibly friendly, cuddly cats. But it's difficult to get a healthy Snake or a cat to do something that they don't want to without an alternative incentive.
There’s also the fact that many cats are pickier about who they get close to, which I see as reflective of Snake Primaries and their vetting process. You have to earn a cat’s affection, just as you do with Snake Primaries. There’s a reason being chosen by a cat feels similar to being chosen by a Snake. On the other hand, for Badgers, there is already a base sense of loyalty to others, even strangers, simply because they are people, which I find is more similar to how dogs latch onto people. A recent study has suggested that the friendliness that dogs exhibit, a key part of their domestication, came from a similar genetic mutation as what causes Williams syndrome in humans, a disease that (to oversimplify) causes people to look at everyone as a friend. Meanwhile, cats domesticated themselves because they recognized that being around humans gave them an advantage: humans store grain, grain attracts rodents, and rodents make a good meal.
And… It’s not what your ask is about, but if you'll indulge me, I'd like to explain my thoughts on the Secondaries. Yes, Snake and Badger were selected as the HP mascots specifically with the Secondaries in mind, but I think I can make a decent argument for Cat and Dog.
Dogs are pack hunters and display alloparenting. That means they work together, specifically that they invest in each other, taking care of each other’s pups and sharing food resources. I’d argue this is similar to how Badger Secondaries invest in people. Also, most dogs exhibit joy in being given a job to do. Their hard work can vary from actual jobs (service dogs, search and rescue, livestock guardian, etc.) to performing the tricks their humans ask of them.
Cats, however, will do what is easiest to get what they want. A study showed that when given the choice between performing a task to get a treat and simply taking the treat, cats chose the latter. This might sound like the obvious choice, but many animals, including canines, prefer performing a task to get their food, a behavior called contrafeeloading. But cats? They are happy to freeload, even if they’re active and intelligent enough to solve the presented food puzzle.
Then there’s the fact that both cats and dogs have shaped their behavior to better communicate with humans. For dogs, these are mainly instinctual behaviors or literal changes to their form, built into their species over thousands of years, like understanding human pointing from birth and evolving expressive eyebrows to better communicate their feelings to humans. Meanwhile, cats learn to continue meowing past kittenhood, depending on their situation, by reading their humans' reactions. Feral cats don’t meow at the same rate as house cats because adult cats generally communicate through smells instead of sounds. Meowing is something cats developed for humans.
Anyway, I understand if you still consider your loyalty to look more dog-like. Perhaps the subtle affections of cats aren't your style. But I hope you enjoy this alternative way of thinking about things!