Sane, signed from Manchester City in July 2020, has struggled to string together a consistent run during his time with the Bundesliga champion.
The flying winger was one of the Premier League's most exciting attackers during his time with City yet found himself a peripheral figure under Hansi Flick last term. On Sunday, he was taken off at half-time after drawing jeers from Bayern's support in Julian Nagelsmann's first competitive home game in charge.
Sane had 28 touches, two attempts – neither of which were on target – and created one chance in the first half against Cologne, with his replacement Jamal Musiala coming on to make an impact by teeing up Robert Lewandowski's opener in an eventual 3-2 win for Bayern.
Rummenigge, who left his role on the Bayern board at the end of last season, believes the Germany international's predicament, and the treatment he received from some Bayern fans, is akin to the reaction Robben initially received.
Robben joined Bayern from Real Madrid in 2009 and went on to become a club legend, but Rummenigge recalled that, the Netherlands attacker once considered leaving.
"At first I didn't like it, I feel sorry for him [Sane]," Rummenigge said on Bild's Reif ist live podcast about what happened on Sunday.
"He tries but has no confidence. He didn't play a good game. With his transfer fee and salary, the criticism of the fans is slowly coming up.
"I remembered 2012 in the stadium. There was the famous home final [in the Champions League]. It reminded me of the sad Arjen Robben. He missed a penalty in the final.
"Arjen was almost ready to ask for his release. But we said: 'No, we won't do that.' He was an important player for Bayern Munich. We supported him.
"Jupp Heynckes built him up as a coach. He was extremely ambitious. The result: a year later he was the match-winner in the Champions League final against Dortmund."
Since starring with a goal and two assists against former club Schalke on his debut, Sane has featured 46 times across all competitions for Bayern, which ranks him third behind stalwarts Manuel Neuer and Thomas Muller (both 48).
However, only 28 of those appearances have been starts, while he has been taken off 22 times – Sane trails only Serge Gnabry (23) and Kingsley Coman (29) in that regard.
Sane has contributed nine goals and eight assists. His tally of 45 chances created is just above former team-mate David Alaba (38) but way behind Muller, who tops the list for Bayern across the period with 125.