I've read that cats cannot live on a vegan diet (they may even die). Is that true? Why is that? Is it the same with dogs?
There are commercially available vegan dog foods that can be safely fed to dogs. Brown et al. tested vegan dog food on a set of Siberian Huskies:
In 2009, Brown and colleagues  reported the results of a study of 12 sprint-racing Siberian Huskies fed either a commercial diet recommended for active dogs (n = 6), or a meat-free diet formulated to the same nutrient specifications (n = 6). The commercial diet contained 43% poultry meal, which was replaced by maize gluten and soybean meal in the meat-free diet. The dogs were fed these diets for 16 weeks, which included 10 weeks of competitive racing. Health checks were conducted by a veterinarian blinded to the dietary regimens. All dogs were assessed as being in excellent physical condition, and none developed anaemia or other detectable health problems.
No such product exists for cats as of now. Theoretically, there is nothing that cats require that can't be obtained form plants or synthesized, but creating such a diet is likely to be quite expensive.
This is my non-expert opinion: dogs and cats have bodies that can process meat, while humans have bodies designed for a plant-based diet. But even cats and dogs can live healthily on a plant-based diet, simply by ensuring that they get all the necessary types of foods and nutrients, which are all available in the plant kingdom or in synthetic supplements.
I know from personal experience that dogs can live healthily on a veg diet. I once had the pleasure to dog-sit for a couple, friends of mine. They told me that their dog voluntarily turned vegan by refusing to eat the usual (meat-based) dog biscuits! I used to feed him the vegan food that they left prepared and he loved it! He was very active and looked very healthy to me. I have not come to know any vegan cats yet, although I hear that there are many.
Taurine, which is said to be essential for cats, may be found in some species of seaweed or in synthetic form, and a good vegan cat diet must include that. One should research carefully what their other dietary needs are in order to include vegan sources in their food, and watch their health closely with the help of a veterinarian, especially in the beginning. It would be irresponsible otherwise.
If you don't want to prepare your own food you can buy it ready made, which also gives you some assurance that it meets their nutritional needs, but again, please watch their health closely. They're available for both dogs and cats from a quick search, vegancats.com being one example. But I'd still suggest some lovingly homemade foods for the dogs as often as you can.
My dog lives on a vegan diet. I'm buying my dog the Benevo food. My dog likes it and has no problems after two years of eating it.
The company produces both cat and dog food.
That's why I think this is not true. Dogs and cats can live on plants.