Now why is that violin so important to Michiru? This instrument is more than just some ordinary piece of wood; it's a Stradvarius. That name might seem like nothing to you since you probably don't know about Strady and the history of his work. Yet, he is the most celebrated and recognized violin engineer in history.
Antonio Stradivari was born in 1644 to Alessandro Stradivari and Anna Moroni. Most likely between 1667-79, he served as an apprentice to Niccolo Amati in his workshop. But Strad was not one to stay in such a position. He added a bit of 
originality to Amati's design by improving arching, variations of thickness, and scroll shape. The public assume his peak years were between 1698 and 1725 when he established a shop in Cremona, Italy as an Italian luthier (string instrument maker).[1]
He stayed true to his work until his death in 1737. For many years after, though, his geometric designs for the violin inspired the accepted violin model and launched his fame for many years to come. But this man has not just limited to violin. No, no! He was also a sculptor of other popular string instruments such as the harp, guitar, and cello. It is estimated by the Smithsonian that only 650 of his 1100 or so works still survive.[2] Bids for these violins rank in the millions!
Many knock-offs were created after his death in tribute to his works and his skill. The Smithsonian reports that "the presence of a Stradivarius label in a violin has no bearing on whether the instrument is a genuine work of Stardivari himself." Many people are tricked into buying one of "his works" because of the lack of authentification.[2]
Obviously, Michiru's violin is much more important than it seems. Only a few professional and virtuosi musicians own a Strady. Such an exclusive list includes the Davidov Strad owned by Yo-Yo Ma, Mstislav Rostropovich's Duport strad cello, and most-recognized and high-prized Soil Strad of 1714 owned by the prestigious Itzhak Perlman (violinst, like Michi!). Even Sherlock Holmes and Napoleon are rumored to have owned one![1] I'd have to say that I would take very good care of mine and make sure it was always by my side.
But this Stradi violin tells us a lot about Michiru and her character. Now there are two ways to take this. 1) Her background is very rich, so she's been affluent and used to luxury for quite some time. This does give a possible explanation for fans' dislike of her "uppity" attitude and classy wear. Or, (2) Michiru's violin is not a Strad and 
she only said that as a way for her to gain attention. #2 does give some reasoning why Mugen Gakuen would accept Michiru above other applicants; what school wouldn't want an exceptional student AND a Stradivarius? Michi's a very wise girl.
Still, there's a possible third. Michiru's insight may be so strong that she is able to sense the passion and feelings within her violin and recognize it as the work of Stradivarius. For her to have - presumably - little experience in authentication and appraisal, her perception must be highly above-level. Not only that, but Michiru is also able to tap into the strength within the violin and use it for her own purposes, almost as if she's in tune with the emotions of its creator. Yes, this is does sound a bit fanciful and "dreamy," but it fits the magical atmosphere of BSSM.
Whatever the case may be, the Stradivarius violin is nothing to joke about. It is a symbol of wealth, history, talent: all characters of our *lovely* Michiru. Nothing could suit her better than the best this world has to offer.
1 Stradivarius. Wikipedia.org.
1 Stradivarius Violins. Encyclopedia Smithsonian.