What did Peano meant by "R. Universitate"? In a book written by Giuseppe Peano, it is written "Prof in R. Universitate de Torino" i.e. "Prof in R. University of Turin".

I guess "R" stands for "royal" but I have no proof of that.
[EDIT: corrected adding "regia" after @MassimoOrtolano's comment].

Yes, it surely stands for regia or reale (both words mean "of the king" in Italian); that adjective was common in pre-war Italy and there are other "royal" universities around. The standard Italian form is "regia (reale) università di Torino"; Peano wrote in an unusual language.

Would this and this document be proof enough for you that it was the full name?