Beast Factor (獣因子 Kemono Inshi) is a genetic component that allows beastmen to transform into their animal forms in BNA: Brand New Animal.
Description
In addition of endowing a beastmen their animal form, Beast Factor also endows them with superhuman abilities related to their animal form in question. For example, bird beastmen possess wings and talons for flight and grabbing things respectively like birds and rhino beastmen possess enhanced strength and durability in addition of horn to damage enemies with.
The extent of a beastman's shapeshifting ability seems to relative with how they attain this genetic component. Whereas regular beastmen who had it from the beginning mostly limited to a single anthropomorphic animal form (notable exception of this being Shirou who attained extra Beast Factors filled with vestiges, if not literal souls, of his fellow beastmen inhabitants of Nirvasyl, through being submerged into a pit full of their blood and corpses that led to his transcendence into "transcendence" into the original Ginrou), those who received beast factor through artificial means like Nazuna Hiwatashi and Michiru Kagemori can freely assume any animal form they wish, if not downright copying it, including a full animal form. As beast factor's condition is relative to the bearer's emotions, they must not succumb to stress to prevent it from consuming them, a condition that referred as Nirvasyl Syndrome.
Biological or Supernatural?
Shirou's violent transcendence into Ginrou, an event that hinted beastmen's supernatural nature
Although it was widely believed that beastmen are a biological species even by Barbara Rose, unique powers displayed by Shirou, Nazuna, Alan, and Michiru showed that beast factor possesses supernatural elements, a fact purebred beastmen are aware of. According to Alan, a beastman may transcend into a god-like entity by being ritualistically bathed in blood gathered from a thousand beastmen of the same species (in his case, wolf), though accidental exposure will do the trick even if the blood in question being taken from multiple beastmen species at the same time. The latter method of transcendence however, had a side effect where the recipient's body will house vestiges, if not actual souls of the donors of the utilized blood via. beast factor within. Whereas such downside explicitly viewed as an affront by purebred beastmen through Alan's expression during their final battle, Shirou argued it as a gift since many of beastmen vestiges he carried came from those of his late friends and loved ones who perished during Nirvasyl massacre, encouraging him not to give up in his campaign to save his fellow beastmen and thus lived up expectations from being the legendary Ginrou and a protector. In the case of human-turned beastmen who received their beast factor through a botched blood transfusion on the other hand, they displayed abilities that not associated with any known species, such as extending and enlarging one's limbs at will, generating blinding light through excess energy, mimicking/copying virtually every single one of beastmen's known powers including those of transcendent ones, and other unknown abilities. From this, it's plausible to assume that beastmen were the result of evolution controlled by an unseen higher beings.
It is possible the primal beastmen could pull off similar feats. According to the Major Rose, the original beastmen apparently attained their animal form from animalistic deities they worshipped, presumably out of psychological desire to be closer to deities in question thus first generations of beastmen were originally shapeshifters with no definite form. Assuming it to be the case, said deities may very well the one who created them.