England meet the Netherlands in the Nations League semi-finals on June 6 for the right to face either European champions Portugal or Switzerland in the final three days later in Porto.
"To have a chance to win a trophy in an England shirt does not happen very often," the striker told reporters. "If we win that trophy, in my eyes, it's a better year than 2018."
Kane is preparing for England's opening Euro 2020 qualifiers against the Czech Republic on Friday and Montenegro on Monday.
England topped a Nations League qualifying group that included 2010 world champions Spain and 2018 runners-up Croatia.
Last year they reached the World Cup semis for the first time since 1990 before losing 2-1 to Croatia after extra time.
"Hopefully, we can win some silverware. Of course, 2018 was fantastic but, for me, the main thing is the team and we want to win things. We want to make fans happy and be proud," said Kane, who won the golden boot in Russia as top scorer with six goals.
"People now expect us to do well and before the World Cup there was no expectation, so it's great that we've turned that around," Kane was quoted as saying by the BBC.
"We're the only team from the World Cup semi-finals to have got through their Nations League group, and we had probably one of the hardest groups.
"That was big for us, to prove that it wasn't just a one-off, that this team were really building for something special."
(Writing by Ken Ferris; Editing by Greg Stutchbury)
