My pet/mount suggestion will be a mix of my 2 favourite species:
Bunnies + Dragons!
Appearance:
I suggest a mammal Dragon Bunny with pink and brown stripes (the colour of stripes could be changed) in a tiger pattern with no wings.
It could have a medium mane in the neck, wrists, ankles and tail, and the tail will be long with the bunny “pom pom” at its end. Its eyes are big and can change colours to show their current emotion, their brighter stripes could glow, and males and females have pouches in their bellies to carry their eggs, food, toys... and have a great sense of smell and vision, and of course, hearing.
It is a very intelligent top omnivore predator (one of the top from its world) that can run at high speed and jump very high and very far, so it needs strong paws and claws, they could live in almost everywhere, since holes or caves, or at tree tops, running and jumping from tree to tree, to open spaces.
Special Powers:
They have elemental powers of many kinds, but while they could have them all, they only choose one to master the rest of their life at the age of 6, while almost ignoring all others.
They use sound, light, electromagnetic fields, fire, ice, space, toxins… to communicate, orientation, defence, attack, localization…
They look all with the same appearance, but if people look carefully, they could see which is the elemental power of a certain creature, like some of the examples (all names are WiP because I’m really bad giving names to thinks, like I didn't even name this Dragon Bunny creatures):
-“Burners” are fire manipulators that are normally burning or smoking something (and activating fire alarms…), they like to swim in volcanoes, and their owners should have fireproof carpets and furniture and a lot of fire extinguishers at hand.
-“Jumpers” are space manipulators that can create portals or teleporting, they normally get stuck inside small areas where they teleport into for curiosity (like cars or wardrobes, and they sometimes teleport to the school or work place of their owners looking for help because they got their head stuck in their owner underwear drawer…). They also have nearby open portals to the nearest food source (that normally means the someone’s fridge…).
-“Singers” are sound manipulators that have the ability to not shut up (but they can shut up others by creating inverted sounds), they are always making sounds, singing melodies… and not only sing with their mouths but also by vibrating their body and fur. They can even learn to talk human speech, and since they don’t know what is lying or keeping secrets, they are regularly saying embarrassing things of their owners (with pride in their voices) to everyone who is listening (like friends, family or the person we have a crush…).
-…
The variety and strength of elemental powers from those creatures depends from their age, experience and intelligence, an example of elemental powers of a “singer”:
By using and manipulate sound, they could make horrible noises to make enemies go away, or when attacking can calculate the rupture vibration of their targets to make their heads or bodies explode. They can produce loud sounds (in infrasound or ultrasound range) to communicate long distances, and beautiful sounds and melodies to socialize, relax or play, they can also use echolocation, hard sounds to make an armour of horns, spikes and plates, and have a very accurate aim to shoot enemies in the middle of friends.
Growing and Reproduction:
They grow fast, their eggs are round and at the size of a handball ball and while their unfertilized eggs are fragile with dark colours, their fertilized eggs are tougher with a glowing bright light. They reach adulthood at around 13 human years with the size of a horse, and can double it for their max size at 30, and live for thousands of years.
They normally live in groups with different species, and since they prioritize being happy and then making others happy, they often prioritize their old social group over family or even recent mates, but sometimes, when unhappy, they leave to travel alone, with their social group or old mate to find a new territory or a better group.
Females lay eggs 6 times per year (every 2 months or around 8 weeks) and lay 2-12 eggs at a time (every time in an even number, with a mate or not, and even when they have a mate its usual that half of the eggs are unfertilized), one week before next mating season (each full season is 2 months, that means 1 month and 2 weeks (6 weeks) to test a new mate if they don’t have one, 1 week to get pregnant and lay the eggs, and 1 week to hide their eggs), that each parent hide half of them in their own territory (it’s an excuse for egg hunt, just don’t break a glowing egg or you will die by their parents, they will know…).
Both male and female can produce milk, if they are mating or are in large groups (not only to feed their and other babies but also each other as a socialization feature), that is too strong for humans in its pure state (needs 10ml of milk in 1l of water to be drinkable, and even that is strong), like their unfertilized eggs (5ml of mixed egg in 1l of water), but are very tasty. In some groups only those who are producing milk hunt and feed the rest of the group with their milk and unfertilized eggs.
They are monogamous, and only mate and lay fertilized eggs if both mates were together during an entire full season, while they test and tease each other trying to make the other quit (if they like each other), after that, they stay with each other while they are together, and play and mate a lot, even outside of the egg lay week.
Lore:
These creatures are one of the most and less aggressive ones at the same time in Gatestriders home planet, since they are the one of the top predators, they don’t fear new creatures, in fact they are very curious.
When Gatestriders arrived the first time, they didn’t understand those creatures, other predators avoided them (or hunting them…), but they seemed very peaceful, they only like to play, follow people, imitate them, and steal their stuff (and sometimes eating their cars if some Gatestrider forgot their lunch inside of it). Until one day someone had the really bad idea of killing one, maybe to try if they taste good or for revenge for their missing car. When the army arrived, the entire village had disappeared from the map, they still tried to hunt them down but Gatestriders had so much casualties that they decided to try other approaches.
The family’s groups of those creatures had a lot of other species and kinds, and they easily accept new faces that prove that they are friendly and work for their community, what made that some scientists and their Thmprs were accepted in their groups while studying them. The problem is that they don’t like to be trained because they don’t want to be part of the Gatestriders society, but even that somewhat changed...
One day, a kid found an hidden glowing ball and took it with her, it hatched in her house and they easily become friends, it gave a lot of work taking care of it but she, with the help of their family, could teach and train it how to behave inside an house (more or less), what it should eat (or steal) or not, and living in society (or almost). The creature was so smart that in few years learned to understand human speech, and even learned reading books (when it didn’t eat them), and then it taught to some wild ones of its kind some of what it had learned that taught to others and others...
They quickly became very popular between Gatestriders, but only few families had some of them because of how much care these creatures need to be happy (and if they are unhappy they just leave), and they are great with kids (except when some family needs to move away because their kid bullied other that had one of those creatures as a pet).
Gatestriders and Reapers:
The Gatestriders like them as pets and mounts because when they are happy, they can glow in beautiful colours and sing wonderful melodies, they love to help and protect their family (of any kind) and can carry a lot of height.
The cons are that they need a lot of space and exercise, because they are a little hyperactive, they can rest but don’t need to sleep, and have a very efficient digestive system (that means almost everything can go in but almost nothing go out), they can’t get fat (that means if they eat too much or too much calories, they will spend all night running around your house or making tunnels below it). They need to be trained by its new family almost after it born to make bonds and create their social group, and they need to have some wealth and insurance because the trouble they normally give with their elemental powers and behaviour.
The Reapers like them as pets because they can use their senses to locate rare resources (living scan hammer…), power some tools with elemental powers, and while Reapers are thumping, they can protect the site from local wild life.
The cons are that they need constant surveillance to prevent them of making tunnels below the mining site, not eating expensive things (like equipment or the rare resources they are looking for) or eating too much.