The quote is attributed to him in many many animal rights banners. Is this attribution accepted by historians or philosophers?
I found the sentence whose Greek was referenced in comments in the OP.
In Regimen, XXIV,
The soul of man, as I have already said, being a blend of fire and water, and the parts of man, enter into every animal that breathes, and in particular into every man, whether young or old. But it does not grow equally in all;
Wikipedia lists this book as being "Works of the age or spirit of Hippocrates". Note that,
The Hippocratic Corpus (Latin: Corpus Hippocraticum), or Hippocratic Collection, is a collection of around 60 early Ancient Greek medical works strongly associated with the physician Hippocrates and his teachings. They are, however, varied in content, age and style, and are largely of unknown authorship.
- Hippocrates, or someone very like him and at about the same time, did say that
- The English translation cited in the OP is concise (abbreviated) and quite loose.
Probably not, but I can't say this conclusively.
It appeared without attribution on the internet in 1997 on an anti-veal-consumption web site [beware: gross images are displayed with the intent to disgust. The quote is in a 1997-style
<MARQUEE>. The font is all centred and with a black background. You have been warned.] _http://www.tonglen.oceandrop.org/Truth_Behind_Veal.htm
Animals are not ours to eat, wear, experiment on or use for entertainment.' ................ Ingrid E Newkirk [PETA]..................."We have enslaved the rest of animal creation and have treated our distant cousins in fur and feathers so badly that beyond doubt, if they were to formulate a religion, they would depict the Devil in human form." ................William Ralph Inge......................"Look Deep Into The Eyes Of Any Animal, & Then For A moment, Trade Places, Their Life Becomes As Precious As Yours & You Become As Vulnerable As They. Now Smile If You Believe All Animals Deserve Our Respect & Our Protection, For In Alot Of Ways, THEY ARE US, & WE ARE THEM." '.' "The Soul Is The Same In All Living Creatures, Although The Body Of Each Is Different."............................May all beings be free of suffering...................The best way to help these animals is to stop supporting industries that regard animals as food machines instead of as living beings with feelings, wants, and needs................I am in favour of animal rights as well as human rights, that is the way of a whole human being.............Lincoln.................The greatness of a nation and it's moral progress can be judged by the way it treats it's animals......... M Gandhi........... Animals of the world exist for their own reason. They were not made for humans, anymore than black people were made for whites or women for men..................Alice Walker...... .s
From there it bounced off in a couple of places, without attribution, and it was finally attributed to Hippocrates in 1999 on this website [beware: 1999-style embedded music player will fire upon load):
"The soul is the same in all living creatures, although the body of each is different." ~ Hippocrates ~
So in conclusion, it seems to be a PETA-style slogan that was misattributed to Hippocrates. But I might be wrong here. In any case there's no trace of support from philosophers or historians.
ChrisW found something very similar in Regimen, XXIV. So he clearly said something alike, good job! I'm leaving this answer as it explains the overall internet history of the quote. As a side note, Hippocrates was talking about aging, probably not what the current use of the quote is meant to exemplify.
I asked PETA where they got the quote, and they told me it is from this book, published in 1852.
A Discourse on the Times, Character and Writings of Hippocrates. Read Before the Trustees, Faculty and Medical Class of the College of Physicians and Surgeons, at the Opening of the Term of 1852-3.
FIRST EDITION PAPERBACK 1852 · New-York: by BARTLETT, ELISHA.