I've been on a pets kick, it's sort of underappreciated for fantasy despite a lot of players keeping one. Granted most are meant for combat but I know a few people who've had PCs with just normal pets they like. So have a couple ideas:
Silk Moths
Since irl silk moths have been domesticated to be all fluffy and unable to fly let's take this to an extreme. People breed them into all sorts of sizes and colors, different breeds exist based on their size and wing shapes. The most common are the Housemoths, about the size of a small cat with short rounded wings, most have eye spots and come in brown, gold, yellow, or amber colors with white and black ones being rare. They're raised much like chickens and can clumsily glide, be taught some tricks, and are very mild mannered.
Luxury breeds of housemoth have more exotic colors and distinct patterns, some ancient households have bred a similar pattern to their family crest onto the wings of their pets.
Garden Golems
Made by the dwarves, tiny golems which range in size from about handheld to as large as a big dog. Less animal and more semi sentient animate object, they're popular for people who enjoy quiet companions with little need for care or attention, or those who enjoy gardening as the golems tend to grow moss. Some are made to even accommodate planting to serve as walking flower pots.
While they're not very cuddly pets they are great decorations, are easy to care for, allow a lot of creativity to go into 'dressing' them, and since they are semi sentient they an love you more than a normal pet rock. As they're gentle by nature it's a great first pet for kids as small ones can't really hurt them or be hurt by them while still having the needs to teach children good pet keeping practices. Not to mention the smaller ones can be very cute when they get affectionate, plucking small flowers or collecting baubles to show appreciation.