Depends on what animals you have in mind.
Carnivorous animals would of course have no problem at all with protein intake.
But the thing is, especially with herbivorous or perhaps even omnivorous animals, you have to remember that muscle is muscle, and no matter what kind of stuff you eat, your muscle doesn't change its composition - think in terms of amino acid profile. Muscle mass may decrease due to a bad diet, but whatever remaining muscle tissue is left is still composed of the same proteins. You can imagine that these herbivorous animals eat a lot but don't exactly have rippling muscles.
Also take note that most animals aren't exactly "muscle bound" as you would expect. We eat their muscle tissue, which isn't really a lot. And those that we normally eat, like chickens or pigs, are raised in farms where they are given high-quality feeds to help them grow bigger and fatter. Had they actually lifted weights like us, they could probably put on a lot more muscle tissue.
Another thing is that even plant protein has sufficient amino acids, and they are merely low on certain amino acids, so you absolutely have to eat a ton of them if they were your only food source. And that's exactly what grazing animals do. Add to that the fact that they don't lift weights, and it's not hard to see how they can live like that.
Last of course is evolutionary effect. Over time, everything just gets better and better adapted to a certain way of living. Carnivorous animals, for example, usually have a larger more efficient liver (or livers, some may have more) to accomodate the huge quantities of meat it has to digest. Herbivores and, to a lesser extent also a few omnivores I imagine, are more adapted to eating various plant life and are better at digesting them.