Gareth Southgate's men surpassed expectations at the 2018 tournament in Russia before being beaten by Croatia in the semi-finals.
They went a step further at last year's delayed European Championship, beaten by Italy on penalties in the final.
England face reigning world champions France at Al-Bayt Stadium on Saturday, hoping to reach a third straight major tournament semi-final.
"Everyone was over the moon (in 2018) but whether we had the full belief to go and win the tournament, I'm not quite sure," Kane said on the eve of the mouthwatering match.
"I feel like the belief has been building and building over the last four or five years now. We entered this tournament hoping and believing that we can win it.
"We've got to a stage (quarter-finals) now where we expected to be and it's about trying to push on.... We go into that game with maybe more belief than we did back in 2018."
England have not won a major trophy since their World Cup triumph on home soil in 1966.
"We're not here just to reach the quarter-finals," said Kane.
"Tomorrow isn't the World Cup final, it's another step towards trying to win the World Cup. We have to be fully focused on that and try to get to the next round."
No team have scored more than England's 12 goals in the World Cup so far, but France star Kylian Mbappe is leading the race for the Golden Boot, with five goals.
Much of the build-up to the game has centred on his battle with England right-back Kyle Walker, who only returned from injury during the group stage.
"We always wanted to give Kyle the opportunity to be fit for the tournament," said manager Southgate.
"He's been a very important player for us over the last six years -- Kyle is ready.
"We're talking about one of the opposition (Mbappe) but we're playing France, who are a fantastic team, and we're aware of all of their attacking players."
