Rodgers was sacked by Liverpool in October 2015 amid criticism that the Northern Irishman had been unable to address a leaky defence, which continued to cost the team points.
Klopp led the Reds to a fourth-place finish in the Premier League table last season, but question marks still hang over Liverpool's defensive unit, which has already shipped nine goals in the league this term.
"Yes, he got them back into the Champions League with last season's fourth place, which will have bought him plenty of boardroom bonus points," Shearer wrote in his column for The Sun.
"The truth, however, is that Liverpool are no different under boss Klopp than they were under Brendan Rodgers. Brilliant going forward but not so clever at the back."
Liverpool are without a win in their last three games in all competitions, and were held to a 1-1 draw by Burnley in the Premier League on Saturday afternoon.