Homosexuality has been observed in just about every species it has been looked for. Transgender animals may not be widely known, but that may just be a matter of less interest in looking for them and/or writing about them.
There are certainly some animal examples that represent various forms of transgender animals in nature. One is the California Sheephead (Semicossyphus pulcher) which is a fish that is born female, and then can become male later in life. Another example is found in the Green Frog (Rana clamitans) that reverses sex in response to various external factors, and this has been observed in other amphibians as well. But both of these examples are in lineages of animals that are pretty distantly related to us humans.

Photo of two lions one is male sex (left), one is female sex (right), but both display male characteristics.
Well, research published a couple of years ago in the African Journal of Ecology is an example much closer to us. In a paper by Gilfillan et al. from the University of Sussex, five lionesses in Botswana have been observed to grow manes, regularly roar and scent mark, mount other females, and display other very male-like behaviors such as killing the cubs of rival prides which females lions just about never do but is very common for male lions.
This is a mammal we are talking about.
Lions that are female in their reproductive organs, but male in behavior and in some aspects of their outward appearance.
In the wild. In nature.
So, do you still think being transgender is unnatural?