While Lizt was definitely an extraordinary person in the meaning that he was talented and beyond what was normal in his musical talent, that is not how the word is used here.
Extraordinary can also mean performing a job and it is “outside of or additional to the ordinary staff; having a special, often temporary task or responsibility”. This is not one of the more common uses of the word, it will be seen more often in official titles than in everyday use.
The Grand Duchess Maria Pavlovna was the wife of the Grand Duke of Weimar, and she was a patron of the arts. She did support a number of talented artists, including Lizt. While there was likely an official court musician, appointing someone of exceptional talent to an “Extraordinary” position, provided them an income to support themselves while they devoted themselves to their art.