Just a quick correction: Austria-Hungary formed in 1867. Mozart lived in the Habsburg Empire, which was reformed to the Austrian Empire in 1804 and then to A-H in 1867.

And I doubt there's a special name for that, looks like an early design of a frock to me.

It's more or less a regular formal wool coat with heavy embroidery, as was style at that time. The fashionable male colours then were primarily red or deep blue so you'll naturally find many depictions of coats in those colours as it is something a gentleman would wear to a formal occasion such as having his portrait done. I'm not aware of any special names.

Also worth noting that that specific and famous painting of Mozart was actually produced by Austrian paintress Barbara Krafft in 1819, 28 years after Mozart's death.

I think it's generally called Justaucorps.