After searching around for the source, it seems that I read this claim in the back cover blurb of my copy of Peter Scharf's Ramopakhyana - The Story of Rama in the Mahabharata: A Sanskrit Independent-Study Reader. This is a generally well respected text for students of Sanskrit, but whether that means there's any validity to what's written on the back cover, I couldn't say. Are there any other sources which back up this claim?
In addition, if it is true, I was wondering what exactly might be meant be "number of manuscripts". Number of surviving manuscripts? Estimated number ever composed? Etc.
There's a definitional problem in the assertion "Sanskrit has 100 times the number of manuscripts of Latin and Ancient Greek combined" that has to be clarified.
Sanskrit is the scriptural language of Buddhism and there are large numbers of manuscripts written in Sanskrit throughout the Buddhist world; Buddhist temples in Japan, for example, have libraries of sanskrit manuscripts, indeed the Japanese still use a Sanskrit script, Siddham, which disappeared everywhere else long ago, for ritual purposes, like tombstones. A very large Buddhist manuscript library was discovered in the "library cave" at Dunhuan on the Silk Road in China . . . there are many thousands of Sanskrit manuscripts or fragments; perhaps more in that one cave than we have from Greece or Rome; the manuscripts date from the 5th to the 10th centuries CE, mostly.
By the same token, the Catholic Church used Latin as its language of administration -- and still does. When there's a Papal Encyclical today, the "original version" is in Latin, although the Mass is now celebrated in the local vernacular. Additionally, Latin was the language of science and scholarship into the 19th century; Newton wrote in Latin, for example. In places like Sweden and the Netherlands, Latin held on for a long time as an academic language-- but I'm guessing you're not counting Linneaus as a "Latin Manuscript".
So the question is "what number are you comparing to what number?" The number of _ancient_ Greek and Latin manuscripts is very few. The number of _ancient_ Sanskrit manuscripts is probably somewhat larger, but without knowing exactly what you're comparing, it's hard to say. As you note, even the meaning of the term "number of manuscripts" is fuzzy-- do we count different versions of the same manuscript as "one manuscript" or several?
As a very broad guess, if we're speaking of "ancient" manuscripts, older than say, 1500 years, I'd expect that we have many more Sanskrit than Greek and Latin, perhaps a hundred times more? I might add that we also have far more materials in cuneiform than we have from the Greeks and Romans; somewhere in excess of a million tablets are in museums and libraries, none younger than 2500 years, most of them have never been read by anyone since they were made. The British Museum which has 130,000 tablets for example, and while that's one of the largest collections, there are a _lot_ of them around the world. By contrast, there's only one library from Rome that survived complete -- but not intact. It's at Herculaneum, and consisted of some 1800 papyri, carbonized by the eruption of Vesuvius, but sometimes still legible with a lot of clever technology.
Sources
Sanskrit Manuscripts in the Cambridge Digital Library
Copying for the Kami: The Manuscript Set of the Buddhist Canon held by Matsuno’o Shrine
The claim comes from this paper https://pdfs.semanticscholar.org/9e00/c37b1c18726f72392d3886f8404488130861.pdf
Check resources they might have given some proof.