Let me first start by asserting that yoga is an Asian tradition. More specifically, yoga is a tradition from India. More specifically still, it originated out of meditative practices in Hinduism and is rooted in Hindu philosophy and mysticism. Everything from it being cited in many ancient Hindu texts (like the Bhagavad Gita) to its inclusion of mystical ideas like chakras all point it it being (A) Eastern (i.e., from Asia), (B) Indian, and (C) Hindu.
So why is it being so tied to Chinese?
I recently stopped in at the Ann Arbor People's Food Co-op and found a flyer for the Yoga House: a lime green paper with "Yoga House" in prominent bold letters above the Chinese character (or Japanese kanji character) for "pleasant" or "luxurious" (愉). Looking through the flyer, I was initially unable to find any link between yoga and countries in which Chinese characters are used. I thought that maybe the founder of Yoga House was Chinese, but looking on the "About" page, I see "Your Instructor is Michelle Bond" written above a photo of an obviously not-Chinese woman. (Okay, she is a black belt holder in Tae Kwon Do, but that doesn't -- to me -- necessarily explain the China-India equivalence.) I thought that maybe the meaning of Yoga translated to something tlike "pleasant" or "luxurious", but looking in a few places, I find that it translates to "control", "join", or "conjunction" (which would be something like 管, 合, or 結合, respectively). However, 愉 doesn't mean any of the things that yoga apparently does. Nor does the material on the Yoga House webpage point to anything about "luxury" (although it does sound quite pleasant).
And the Yoga House case isn't a unique example of this phenomenon. Looking around at the Yoga section of my local Borders, I found a few books and DVDs that prominently featured Chinese American yogi, Rodney Yee. Now, most of the books on sale showed white men and women doing yoga of varying degrees of difficulty, and I'm not saying that yoga hasn't become a lot more "mainstream" in the US (and becoming Americanized, too). However, there is a difference (at least in my mind) of transliterating "योग" into "yoga" to an audience that reads a Romanized language (and not a Sanskrit language) and casually mixing mysticisms. (Note, though, that I'm not against mixing mysticisms, so long as it is clearly stated why one is doing so; doing it willy-nilly is problematic because it glosses over and ignores the cultural and historical differences between them.)
By the way, if you go to "yoga" on Wikipedia, and then click over to the Chinese language page, the word for "yoga" in Chinese is "瑜伽" (in Japanese, it is transliterated as either "ヨーガ" or "ヨガ"; in Korean, it is "요가"). One meaning of 瑜 is "flawless gem", and (somewhat fittingly) 伽 is used as a verbal designation of the Sanskrit sound "gha". (I am often surprised at how well Chinese transliteration does in being able to combine a rough verbal transliteration of a word with an appropriate -- sometimes poetically so -- definition in the characters used to make up the word.) There is no inherent meaning behind the transliteration characters in Japanese and Korean, as these two languages use a phonetic alphabet that -- like the Latin-based alphabet used in English -- doesn't inherent meanings in each character.
Finally, though, looking through a list of Chinese characters with the same Pinyin transliteration of 瑜, you can find 愉. Yes, they are homophones of each other. However, they also share the same sound as other characters, with a variety of meanings, including "blocking the breath" (not really what one wants to do in yoga),"the mouth of a fish gulping for air" (again not really useful as an image in yoga), "stupid, doltish" (not a good way of advertising), "a military flag" (not really synonymous with the American ties of yoga with peace), and a host of other characters.
Perhaps Yoga House was not aware of the "official" transliteration of yoga into Chinese; or perhaps they chose to just use their own. However, I would argue that most people would not be aware of the above (possible) link between the Chinese character used by Yoga House and the Hindu mysticism-based practice that they teach there. And perhaps most people in the West don't really care. Similarly, I would imagine that most people who did know that it was a Chinese character on the flyer, and that yoga is from Hinduism, would be able to make up some sort of rational justification in their head to make the dissonance fade away. However, I think that there is a kind of "pan-Asianism" that exists in the West (especially in places where Asians are a visual and mental minority), which makes it "okay" to casually blend Eastern cultures in ways that create dissonances for some, but still keep it in the large and diverse bowl that is "Asia." What I think that this sort of thing does is analogous to mixing together the Norse god Odin with the Christian God; Loki with the Devil; Thor with Jesus; Tyr with the archangel Michael; Heimdall with St. Peter; and so forth. I mean, they are both "Western" traditions (and they both have "overlapping" responsibilities and personas), so that's all good, right?
(I wrote about something similar a few years ago about depictions of the Buddha as a fat man in the West.)