The Spurs boss feels that the England captain might not be held in such high regard if he decided to move to another club.
Speaking after Kane scored twice in Tottenham's 3-0 win over Leicester, Mourinho said: "I think everyone thinks he's a fantastic player and Tottenham is so lucky to have him because he's the player, the person and the Tottenham boy.
"All this together makes him really a special player for us that probably wouldn't be so special playing for another team. He's really special for us, he's really special for Tottenham.
"Of course we want him to be happy and for him to be happy he wants victories, score goals, and I'm so happy after an incredibly difficult injury he's coming in the direction where he's going to end the season perfect.
"Hopefully the beginning of next season he will be even better."
The former Manchester United coach added that Kane is "second to none" when compared with some of the forwards he has worked with during his illustrious career.
Mourinho told Sky Sports: "It's difficult for me to compare strikers, but I had almost in every club some of the best, and of course he's second to none.
"He's just different, different qualities. But as a striker, as a goalscorer, as a team player, as a leader, I think he is fantastic.
"It's difficult to make him any better. The better the team is, the better he can be. But it depends more on us, on the team than himself.
"He's an amazing professional, he works very well, he cannot work better than what he does during the week, he's a fantastic striker, I'm so happy and Tottenham are lucky to have him."