Weeds are weeds because they grow where they are not wanted. They might be desirable plants in the wild within their native ranges, or beyond their native ranges where they are useful, but for one reason or another, are undesirable in other situations. In fact, many of the most invasive exotic (non-native) weeds were imported because they were useful for something, and then escaped.
Many invasive exotic weeds that were not imported intentionally by humans likely stowed away intentionally by their own means. Some produce edible fruits that contain their seed so that animals who eat the fruit transport and disperse the seed. When animals such as cattle, swine, sheep, horses and chickens are imported, they can bring such seeds with them, and have already done so.
Not all plants have such mutually beneficial relationships with the animal vectors who transport their seed for them. Rather than expend resources on fruit to appeal to, and reward the animals who eat it, they produce seed structures that cling to animals. Most get tangled in the hair of mammals. Some get wedged into cloven hooves. A few are just sticky enough to stick to the feet of birds.
It is sneaky and exploitative, but effective. Most of these sorts of seed structures stick to the fur only for short distances before falling to the ground, where they really want to be. Some types cling for longer distances, in order to take advantages of larger migratory mammals. Dispersion is their objective. Even though they provide no benefit to their vectors, they do not intend to harm them.
However, they sometimes do. Sharply pointed seed structures that are designed to slip smoothly into fur, but not come out easily, can get into eyes, noses, ears and throats of innocent animals. Foxtails are the most dangerous, and sometimes need to be removed by a veterinarian. Burclovers get tangled in soft fur, and sometimes do so in very uncomfortable clusters.
Domestic dogs and cats are more susceptible to the dangers of weed seeds than wild animals are, because their fur is longer, shaggier, and maybe curlier.