Zidane stepped down as Madrid coach at the end of last season, just a few days after guiding the club to a third successive Champions League title under his watch, FourFourTwo reported.
The former Madrid player had a remarkable impact during his time in the hot seat, also winning a La Liga title in his first full season.
Solari follows in his footsteps after being promoted from Madrid's Castilla team, replacing the sacked Julen Lopetegui.
But the Argentinian does not want comparisons to be made between himself and Zidane.
"Zidane is incomparable with anyone," Solari told reporters in Friday's news conference.
"He is a legend, he had a fantastic level. He was an exquisite player, with whom I was lucky to play.
"But as a coach he will probably be the most successful in the history of Real Madrid. I admire him as a human and we must admire him in his greatness. He is unique."
After initially taking over on an interim basis, Solari earned himself a contract until 2021.
Some have suggested he stands little chance of seeing out his contract given Madrid's history of hastily dismissing coaches, but Solari is unconcerned, acknowledging that everyone is just passing through the club.
"Nothing changes to me, nothing in the 11 years from when I was here before [as a player]," Solari added.
"The important thing is being here and the commitment you make with your work every day.
"We're all just passing through in Madrid, particularly in this job."