(If you watched, go watch! It's a good video)
Which got me thinking: "What traits might we want from an animal, if we could imagine animals as drastically different from what we have now. What might an alien want that is very different from a human?" So lets start with something easy. Everything living needs energy to survive and be active, so it's a reasonable assumption that sentient aliens would need to eat. (Except maybe those fancy green aliens that cheat). While it's true that turning grass into meat is inefficient calorie-wise, it's pretty useful if the grass is useless to us anyway. Some calories is better than none. It's also a big bonus if you get a taste boost in the bargain.
When we look at the chemical factories we call digestive systems in Animals on Earth, we see that specialization is key. More specialization in what we eat means more efficiency. You don't need to worry about specialization when your diet consists of calorie-rich fruit, meat, and vegetables like the diet humans tended to have. Yet, if you want to eat food like grass that is minimal in effective calories, specialization is the only hope. Rather than go through the hassle to turn grass into edible material, easier to have an animal already suited for the job do it for us. Other examples of inedible food include poisonous plants, food waste, and carrion. Perhaps we're looking purely for taste, ala honey (or aphids for ants).
I'd think it a no-brainer that aliens would have some form of domesticated animal that turned inedible food into something edible. The fun part is in imagining how the alien gets the calories from the animal after. On Earth, it's most efficient for animals to produce food continuously, rather than only slaughtering them once they've grown. The big two examples are milk and eggs. Making meat is expensive, it requires the animal spend calories to make the cells, then those cells must be maintained requiring more energy the longer the cell lives before butchering. Comparably, milk and eggs are a single energy expenditure, no energy maintenance required.
Maybe we find an animal that can compromise. Lizards are an animal that can continuously regrow lost limbs, with some even using it as a defense mechanism. Why not a domesticated animal that can consistently grow meat without needing to be butchered? Or get really alien. Have the male be an omnivore similar to humans, while the female has a diet more similar to a horse or perhaps cow. (Or reverse the roles, if that's your preference) Even better if they have some foods to share that both enjoy, such as high-sugar foods like honey. What would society be like if men and women didn't ever share meals, but had the occasional dessert together?
As well as making useful calories, animals produce other useful materials. Bone, fur, feathers, wool, ivory, leather, silk; there's a lot of useful materials we derive from animals, especially before more advanced methods of production are found. We've only recently (relatively) developed alternatives to some of these items that don't require animals to produce. I'd find it incredibly likely that alien civilizations would use animal products as well, in likely similar uses as in our past. Building materials, decoration, and especially temperature regulation are all too good to pass up. Always a great choice in a story when you need a material made is to have it be made by an animal. Which seems sillyat first, but considering that guano (aka bird/bat poop) can be easily refined to become the main ingredient in gunpowder, using animals as chemical factories doesn't seem so odd.
The last major usage for domesticated animals is as workers. Preferably, they have other purposes from those listed above to be as efficient as possible. The horse is the best example. It can be used to carry things we'd rather not carry, including us. It runs at high speeds, faster than we can. Even better, it runs off of food we don't eat, and can convert it to food we can eat. Milk and meat even. Other examples are great: oxen for heavy lifting, cats for killing vermin, dogs for herding and their enhanced senses, and others. Effectively, anything your alien can't do is good grounds for employment of animals, along with anything a sentient doesn't want to do but still needs doing. Humans can certainly pull along wagons, but the oxen whine less. Dogs have different senses than us, making them highly useful for compensating to obtain a fuller range of senses. Presumably, an alien that lacks in good eyesight could have a more literal seeing eye dog as a companion.
Effectively, if you ever find yourself imagining an alien species, try to look too at what domesticated animals they're likely to have. Look for holes in their biology, then think of ways other animals could help fill those holes. Don't forget to add some variation to the animals too. Most farm animals are pretty docile, but horses have an attitude and camels are jerks. Don't horse around by making the animal the same as a cow, except In Space! Unless of course, you're making a joke about how aliens seem to like our cows so much. Please remember to milk that joke as much as possible. Hope you enjoyed the read, it's time for me to get mooving to another project.