Just the same: Danny comes from Daniel, so any nicknames would be Dan, Dani, Danny, Danielito.
dani is sometimes short for daniel
Dan is the third person plural form of the verb "Dar". It means "they give".
Daniel Colletti goes by Dani.
Daniel Bizer goes by Dani.
Daniel Buira goes by Dani.
Daniel Lloyd Davey
Daniel Osvaldo goes by Dani.
Dani Shushan's birth name is Daniel Shushan.
Dani Pedrosa's birth name is Daniel Pedrosa.
Dani Zuniga's birth name is Daniel Zuniga.
Dani Dorf's birth name is Daniel Dorf.