Except that Down syndrome isn't the result of inbreeding. Down syndrome is a very specific genetic disorder.
That's a dumbed-down explanation from a third-rate viral site, which exists solely to generate clicks from Facebook, and not an actual medical or scientific description of the tiger's condition.
People have suggested that Grumpy Cat might have a feline version of Down Syndrome as well.
Edit: It looks like the very definition of Down Syndrome (three copies of the 21st chromosome instead of two) doesn't really allow for an "animal version", since the 21st chromosome is specific to humans.