Another day, another cute story about how dogs can grasp elements of human language and use them to communicate with us.

First, there was Mr Waffles, the Yorkshire terrier that“talks” to his owners by pressing electronic buttons that have been pre-programmed with words and phrases. In one of his videos, viewed more than half a million times, the pint-sized pooch stares defiantly at the camera and responds to an empty packet of treats by pressing the “I don’t give a damn” button. Followed by the “bitch” button. Bad dog, Mr Waffles.

Then, last month, researchers reported that some dogs can mentally categorise their toys depending on their use. “Fetch toys” can be distinguished from “tug of war toys”, and retrieved appropriately.

I’m a dog owner. I fully admit to being one of the third of Britons who would rather talk to their pets than to people. At first glance, the study seems to confirm what I already knew – that my dog understands everything that I say – but while interactions like this tell us something about how smart dogs are, they barely scrape the surface of their communicative abilities.

In recent years, many parallels have been drawn between animal communication and human language. We have learned, for example, that some animals have names that they use for each other. Elephants use particular vocalisations – harmonically rich, low-frequency sounds – to address key members of their group. “Hey, Dave, over here!”

In their early months of life, bottlenose dolphins invent their own unique “signature whistle”, which goes on to become the name that others use to call them. During this same time period, mother dolphins communicate with their youngsters using “motherese”. Just like human mums, they adjust the frequency and pitch of the sounds they make to a create a singsong tone thought to facilitate bonding.

Meanwhile, songbirds have different regional dialects, meaning the sparrows near me sing with the avian equivalent of a Birmingham accent – “All right, bab!” And sperm whales, which use “names” and accents, have recently been shown to have their own phonetic alphabet. The gentle giants communicate with patterns of clicks. By altering their rhythm, tempo, duration and number, simple units of sound can be combined to generate complexity. According to the scientists who discovered the phenomenon, this is similar to the way that humans combine sounds to form words.

Whenever these stories break, they are met with surprise. “Aren’t animals clever?! Who’d have thought it?!” But none of this should be surprising. All of these animals live deeply complex, rich and social lives. Sperm whales, for example, live in tight-knit female-led groups, which sometimes come together to form larger groups known as “vocal clans”. They hunt together, babysit for each other and work together to see off predatory orcas. Of course they have complex communication. They need it for survival.

The mistake, however, is to presume – a priori – that animal communication is anything like ours. All too often we try to crowbar animal communication into a human-centric framework, but it’s like trying to fit a square peg in a round hole. We’ll never solve the puzzle, because there will always be a fundamental mismatch.

Just as a cuttlefish would miss the subtlety of human sarcasm, so too humans are unlikely ever to understand the full spectrum of cuttlefish communication. When they dynamically change the patterning and colour of their skin, it’s fascinating, but alien. Different animals communicate in different ways. Sure they use sound, but they also use colour, smell, electricity, vibrations and the medium of expressive dance. There is an abundance of animal communication that we miss because we don’t have the sensory organs or humility to detect it.

Human language – with names and words and syntax and grammar – is just one solution that one species has evolved to help it navigate the challenges that it faces. Non-human animals live different lives with different struggles. If we really want to grasp what animals are “saying”, we need to understand their communication on their terms, not ours.

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Jane Goodall, who died last week aged 91, realised this when she spent time with the chimpanzees of Tanzania’s Gombe national park. Through careful observation, she uncovered a wealth of previously unrecognised complexity, including the discovery that chimps communicate using body language as well as sound.

Contemporary ethologists have built on this. The work of Cat Hobaiter from the University of St Andrews shows that chimps have a rich repertoire of gestures that they use for close-up communication. They don’t have “words”, but they do have flexible articulate signals that have meaning. So, when a chimp makes an audibly loud, long, slow scratch, it can mean “groom me”, but it can also mean “let’s travel”. And while they may not have specific gestures or calls for different types of food, such as figs or seeds, maybe they don’t need them. What humans and chimps consider useful information is different.

So, instead of marvelling at the ability of pooches to press word buttons, take time instead to marvel at the way dogs communicate with dogs. When my mutt sees his best friend, for a short while his arthritic limbs become jaunty and joyful. There are play bows, tail wags and butt sniffs. It may be no Shakespearean sonnet, but it’s expressive and beautiful in its own right.

• Helen Pilcher is a science writer and the author of Bring Back the King: The New Science of De-Extinction. To support the Guardian, order your copy at guardianbookshop.com. Delivery charges may apply