I have a half-animal, half-human hybrid character who gets indoctrinated into a Nazi-like group of other half-animal hybrids who hate humans and want for them to be extinct.
She's a child when this happens and she later becomes the main antagonist who threatens the whole world yada yada.
What I'm asking is if it's wrong to portray her as a villain because she kinda got brainwashed or if it's okay because she chose to go along with it.
This is where the distinction comes in between a villain and an antagonist. An antagonist is merely someone who opposes the protagonist(s); they do not necessarily have to be villainous. If you're uncomfortable with the idea of her being an outright villain, you can choose to portray her in a more sympathetic light, playing up the fact that she's been manipulated and brainwashed her entire life and genuinely doesn't know any better.
On the other hand, it's perfectly okay for you to portray her as a villain if you want to. If, as you suggested, she "chose to go along with it" and wasn't truly brainwashed at all, I would find it very difficult to have any sympathy for her.
It's up to you to choose how sympathetic or villainous to make her, depending on how you want her story arc to unfold. For example, if the story ends with her realising the group's ideology was wrong and siding with the heroes, then you should make her more sympathetic. Be aware, however, that no matter how you portray her, there will likely be a few readers who interpret her in the opposite direction (i.e. finding her sympathetic even if she isn't meant to be, or vice versa).
There’s really no right or wrong to this. Villainy is not only subjective, but exists in shades of gray. As the author, it is your choice whether to make the character a villain at heart to was exposed to their true nature by a group of like-minded creatures or a kind-hearted character gaslighted by an evil cult into a hatred that ill-befits them.
You choose, although as @F1Krazy said, there will be readers who don’t see it the way you intended. Does the character have a redemption arc? Will the character be killed off by a hero,who then feels guilty about it? Does the character become a leader or recruiter for the group, indoctrinating others as they were indoctrinated?
Plot points like these can help you make the decision.
It depends on your skill level, and what you are trying to do.
Maybe it will be a bit easier to take an example from a bit farther in the past so as to have a little less emotional baggage attached. Consider the ancient Romans, round-about the first century AD. They did lots of things that we today find horrible. Slaves, gladiator contests to the death, animal fighting from roosters fighting roosters up to tigers fighting bears. And some other things that I won't describe here, even in side-ways terms. It is the usual opinion these days that such things are completely horrible and unacceptable.
But consider it from the point of view of a citizen of Rome at the time. Consider a Roman mother trying to teach her children how to behave in the culture of the time. Say a well-to-do Roman, somebody with enough money for a house and some land. If this person's parents were to have trained him from birth to hate slavery, then he would never buy a slave. He would be at a distinct disadvantage in his culture.
So, in a sense, his parents would have been bad parents to teach him that way. They would have been severely poorly preparing him for life. Maybe even condemning him to relative poverty. Maybe, if things got bad enough, he might even be enslaved himself.
The culture around us is a powerful force. Portraying it as such is a challenge. Not portraying it in terms of cliché and surface appearances is a bigger challenge.
So, portraying the NAZIs as half-human half-animal is potentially going to get in the way of telling that story. You are going to have to work very hard to portray the experience of this character through that visage. It's a fairytale surface thing, the big-bad-wolf image. You need to show the character, and in particular the internal reasons for what he does, not just his animal snout. You want to avoid "look at the evil animal."
Or, to put it another way: The thing that made the NAZIs evil wasn't the fact that some of their uniform insignia had skulls on them.
However: If you manage it, you could well have a remarkable work. As an example, consider Maus. This is a graphic novel about NAZI Germany. Jews are depicted as mice, NAZIs as cats. But it manages not to be cliché. And even though it is a cartoon, it manages not to be cartoonish. The phrase is "subverts expectations."
It largely depends on how the indoctrination is done, but from your description, the ideology is learned and the character in question has the ability to question the dogma of it, does so, and is not changed by the decision and it leads her to cause harm. A reader can understand the tragedy of how one event may have led her to this opportunity to go down a more Nobile path, but the point of a villain is to act as the incarnation of immorality in a morality tale (As other answers say, antagonists are not defined by their morality, but rather by obstructing the goals of a hero. For example, in a tale about a man, stranded on a desert island, the seas, storms, lack of civilization are antagonistic forces, but they aren't evil. They are just obstacles to the man's goal to get home.).
One can sympathize with a villain's motivation, even that the villain is a victim of her own society.
It is one thing if the villain's actions were beyond her control, she was puppeted and would not have done them but for a puppet master. But she chose to do her villainous deeds of her own free well. She has seeded the wind, and now she reaps the whirlwind.
Two wrongs do not make a right. Can we empathize with her horrible upbringing? Absolutely? Does it justify her paying evil onto others? No.
One wonderful character to follow with this kind of storyline is the villioness Demona from the Disney Animated Season "Gargoyles," who learns of the fated destruction of her clan of gargoyles by the humans and attempts to save her clan without warning them. However, her plans end up leading to the foretold destruction, fueling her rage against an innocent human, who in turn creates a feud between her and his family that lasts for a millennium. But all of her suffering reinforces her hatred and distrust of humans, which fuel her growing desire to wipe out humanity as a species. But everything that happened to her was because she assumed the worst in others, despite not having complete control of the situation, nor a complete understanding.
