Kane missed a penalty but netted twice to surpass 200 league goals in his career as Spurs maintained their unbeaten start to the season with a 2-0 victory at the City Ground on Sunday.
The England captain joined Andy Cole on 187 Premier League goals. Only Shearer (260) and Wayne Rooney (208) have scored more goals in the competition's history.
"It's still a long way away, for sure, but I still feel fit, I feel I have many more years left in the locker," Kane told media.
"Every season I want to score goals, I want to be amongst those names, and I'm excited for the challenge.
"Alan and Wayne are two of the best strikers England have ever had so to even be in that category is nice for me. But I want to keep pushing myself, see how far I can go. So far, it's been a good start to the season."
Tottenham's third victory in four games left them third in the league standings on 10 points, two behind north London rivals Arsenal.
Kane has scored four goals this season and his form has been key to Tottenham's best start to a Premier League campaign since 2009-10, when they opened with four straight wins.
Last week, Kane set a Premier League record for the most goals for a single club, moving clear of Sergio Aguero's tally for Manchester City.
"Harry has to be our point of reference on the pitch at every moment, he has to be the leader for this team and I think he is working very well," Spurs manager Antonio Conte told reporters.
"He's totally involved in this project, and I am really happy for him."
