Can a I take a willing brother away from an abusive wife?

I think the key phrase here is "take a willing brother away".
What reasons has the daughter been given for not being able to see her father?
There is much missing here. If your brother is suffering from diagnosed dementia and your brother's wife has been made his guardian, then what you are discussing is basically kidnapping your brother from the person he chose to have in his life.
If your brother does NOT, however, have dementia, then he is free to leave his wife at any time, with half the assets of their estate, and live wherever and with whomever he pleases.
It sounds as though you are not in close contact, have not been, and don't know the facts in the case. So the key is firstly to make a family visit, as apparently you plan to do. Do remember, when you discuss with his wife your brother's condition, she is not beholden to share any of their private life with you, and if your brother is demented his description of his current life is not reliable. It will be clear to you when you visit how well he knows you, his reaction to your visit, and his circumstances of living.
As to an allowance, that would be called normal. A demented person cannot reliably keep money, and know what to do or, more importantly what NOT to do with it.
I wish you luck and hope you will update us with what you find on your visit. I hope you will start out by offering your support to your brother's wife as she attempts to care for--if she is correct--a husband who is suffering from dementia. SHE is the lioness at the gate. She is the person your brother chose to live with and love, and SHE is the one who is left now to care for him as he descends into what may be dementia.
As to attorneys they should not at present be brought into the conversation. It is the wife's right to care for her husband, if she is willing and able to do so. She already has conservatorship or guardianship, and you would NEVER be able to rip that from her unless there is outright evidence of abuse.

I am not criticizing here just want to confirm what others have said. You live too far away to make certain assumptions. There are many members here who talked to parents, long distance on the phone thinking everything is fine. Then they move Mom or Dad or both into their home and find that things were a lot worse than they thought. If your brother suffers from Dementia and is Showtiming, then u need to take what he says with a grain of salt. She may just let him have pocket money because he can no longer handle money. Maybe he gives it away, maybe he has been scammed. She does have to protect what is hers too.

"His wife has appointed herself through the state as his caregiver."
I don't understand this either. Is she being paid by Medicaid to care for him or another government agency? That does not give her legal rights. That would be a POA or guardianship. You maybe able to question a POA but guardians are appointed by a Judge. You would have to prove neglect to get that ruling changed. I do think the daughter should be able to see her father. But that's her fight.

You need to visit. I would call APS in his county and ask what rights do u have and could they do a well check. You want to make sure brother is safe and cared for. That the wife is alienating the daughter. You need to realize that a wife trumps siblings. She is also the one caring for him. Its also her house.

If you find u can take him with you, you need to realize as his wife has rights. I really don't see how you could do this without a lawyer. For no other reason to protect what is his for future care.

Have his daughter contact the local police and request a welfare check. His daughter at the very least should be allowed to see her father. As siblings, you're a bit farther down the totem pole in terms of having any power.

Has your brother expressed a desire to leave his wife and live with you? He could very well be 'showtiming' when he talks to you. Many elders with dementia can keep it together for periods of time. Some are so good at it that they put one over on their own doctors.
If he has not been declared legally incompetent and his wife made his conservator/guardian, she does not have the right to decide for him. You may run into trouble if he does have dementia and changes his mind and wants to go back home to his wife. His wife does not sound like a person who will just let him go happily to his family because he wants to. She will call the police and APS (Adult Protective Services) and you could get into trouble.
I would say to consult with an elder law attorney and see what your possible options are if any.
Also, a caregiver (even a spouse) isolating an elderly person from family and friends is considered elder abuse in many states. In some states it is also considered unlawful restraint. An elder law attorney can best advise you. Talk to one.

"His wife has appointed herself through the state as his caregiver." If this means she has guardianship, then she can have you charged with kidnapping if you take your brother anywhere without her approval.

I suggest contacting APS about not being allowed to visit. In at least two cases I know of, APS in East TN checked on an elder and required family to be allowed to visit.

If he is "lucid" and doesn't really have dementia he can remove himself from her clutches at any time he wishes if he isn't physically or medically impaired...so why doesn't he? If he talks to you on the phone he can call 911 just as easily. You won't need a lawyer if he's competent and not impaired in any way.

If he IS cognitively impaired and she is mistreating him you will need proof. You must contact APS for their county -- but don't do this until you are there in person and can assess the situation. If you go there and she blocks you from visiting him, then I would involve APS for sure. Maybe she also has some sort of cognitive problem as well. If your brother does have dementia he may be having delusions, which is a common part of the disease.

Does she have any family that you can talk to? If I were in your situation I would have someone go there to make in-person contact with him. They can "test" his memory by asking him very specific questions (what day it is? what season? Who is the President? What is your address? etc). My MIL could carry on a very breezy conversation as long as you asked her vague questions (How's it going? Whatcha been up to? Is everything alright?) This is called Apparent Competency. This is why you need to ask very specific questions about current things.

FYI pursuing guardianship through the courts can be very expensive ($10K is a number often mentioned on this site) and if the judge doesn't like the family fighting may appoint a neutral (and unrelated) 3rd party as guardian. Good luck!

There is a really BIG chance you have a very limited and probably unrealistic understanding of his real condition. You talk to him briefly and he seems ok...look up Showtiming. It basically means being able to make a good impression for a brief time. People on this forum report their loved ones doing this all the time. If he really is okay why doesn't he leave the house, or need a caregiver, or be given a $20 allowance? It makes no sense. Does he have some disability or advanced age other than the dementia you seem to doubt? Many posts on this site involve long distance interference in someone's care because they think that they know better than the person actually doing it 24/7. Tread carefully here....you may be making some incorrect assumptions.