I'm not a native speaker of Chinese, but I should know enough to answer your question.
However, I'd have to see the panda's name written in Chinese characters for me to be able to answer your question, and the website you provided the link to doesn't seem to display the panda's name in Chinese text. In Caroline's case, her "Yuanyuan" is each comprised of the same character (pronounced as "yuan") which means "round". Without seeing the written characters of the panda's name, the panda's "Yuanyuan" could have various other meanings besides "round" (for example, "garden"). A word with the same sound (in this case, "yuan") can have many different meanings depending on the tone it's pronounced in. Heck, even words with identical tones of pronunciation (there are four in Mandarin Chinese) that sound the same can sometimes have different meanings! You can see why viewing the words written in characters is occasionally necessary to differentiate similar sounding words from each other.
In all likelihood, the panda's name ("Yuanyuan") does mean "round" as it fits the animal so well. Each "yuan" in the panda's name is probably the same variant as well, seeing as it is common practice for the Chinese to repeat the same word when following naming conventions (this is especially true for affectionate nicknames).
Are you a big panda fan? I've been following panda breeding news for a while. The panda population in zoos is booming. There are currently over 200 captivated pandas, mostly in China thanks to the development of panda breeding & feeding technique.