Kane never wavered in his desire to keep improving after becoming the Bundesliga's record signing. He spoke recently about studying the game's greats and having the drive to consistently be one of the world's best.
He has dealt with the pressure of essentially replacing Robert Lewandowski, who left in 2022 as Bayern's second all-time scorer behind Gerd Muller.
"Bayern Munich has a huge tradition of number nines," explained Klinsmann.
"There is always a special eye on them. To fill that spot is just an honour. It's something very special. He knows that he is followed closely by everyone."
And not just in Munich - Khalil said rival fans see it as a privilege to have Kane in Germany: "When the captain of the Three Lions is playing in the Bundesliga - wow, that's big!"
Yet there is no superstar ego on boy from east London.
He is regarded as incredibly professional and a leader among the squad - a player a lot of his younger team-mates look up to and someone who is very respectful, whether that be in training or completing the many media and partner obligations that come with representing a European giant like Bayern.
"He has a royal approach," added Khalil. "When he enters the stadium, he greets everybody, he is very friendly. If there is a moment he doesn't want to talk in the mixed zone, then he excuses himself. A lot of other players walk through with headphones on or use the telephone and try not to look in the eyes of the media.
"Normally he stops, he is always ready to talk. He is an absolutely leading person in that club. He gives a lot of that James Bond gratitude - he is the striker with the licence to score!"
Kane has thrown himself into life in Munich, embracing the culture, trips to Oktoberfest, visiting supporters' clubs and taking German lessons, even if he has previously joked his kids will learn the language before him.
"What he's done extremely well is adapt to Bavaria with so much ease," said Klinsmann. "He became part of the lifestyle, the city and his family just fit into the way of life there.
"It's an international place, Munich, so it's not too problematic but still, you've got to go to another country and keep your work-rate at the highest level while figuring out that your family are comfortable and having a good time."
He is popular, too. Kane quickly struck up a friendship with Bayern's all-time leading appearance-maker Thomas Muller, who will leave at the end of the season, and is known to play golf with the Germany forward and midfielder Konrad Laimer, among others.
Naturally, he is also good friends with former Spurs team-mate Eric Dier, another player well-liked throughout the squad, with both England internationals regarded as being important to team chemistry at Bayern.
"The people love him," added Klinsmann. "He is just himself. He is humble and down to earth. He focuses on what is important and for him, that is scoring goals."