You aren’t getting anything similar because your opponent played e4.
I love the Dutch. But what do I play against 1. E4?

You aren’t getting anything similar because your opponent played e4.

Learn to play the defense that annoys you the most when you play e4. That way you'll learn to play against it better.

What Dutch do you play? There are 3 variations of it.
The classical.
Good choice!
So you're an aggressive player. I'd tell you to play the Sicilian, but the sidelines in this opening, are closed to be main lines - they're very strong - Alapin, Smith Morra [if you accept the gambit], Grand Prix setup, McDonnell Attack, and most annoying - The Deferred Alapin. But if you think your opponent will play the Open Sicilian, be my guest - the Pin variation will give you a better endgame, simply put. and the Kan variation will give you a fun attacking middlegame with many tricks of your queen.
So, I suggest the French Defense. or 1...e5 if you've got a huge memory [= if you're somewhat young = below 35] - e5 is objectively the best move.
But! There is some kind of Pirc Defense which goes like in the following diagram, I don't know what's it called, but I'm having Real problems against it.
I just threw nonsense moves for White, it's Black's setup that matters. it Looks strange, but it's strong, you have attacks I believe. but... it's better to learn it from someone who at least knows how it's called. I'm only facing this annoying thing - never heard of it.
Generally speaking. e4 players hate to face the French Defense [they needs to know too many variations to play it right, so they play the Exchange French - which is an insulting draw, or the Advanced French - which is weak. at least in My level it's what I always see] The French is solid too - he can't 'just crush you' if you know some theory.
d4 players hate to face mostly the Benko Gambit, as a recent chess.com forum poll has shown.

But if you think your opponent will play the Open Sicilian, be my guest - the Pin variation will give you a better endgame, simply put.
And even more simply put, the Pin variation will give you a highly problematic middlegame.
It is a line to play once a Blue Moon against weak opposition- objectively Black is worse, UNLESS he follows the lines suggested by GM Seirawan in his youtube lecture.
Following these lines, he is lost.
cool!
since you're already here [and are an IM], can you please tell me what is that Pirc var. I was talking about? and, what is a Blue Moon?
Thanks in advance, Pikachu.

ah, you meant it should be used only one time. okay cool.
Once in a Blue Moon means extremely rarely, and it's about the Pin variation, not Pirc.
Very well sir, thx for your time :-)
I love the Dutch - it's off-beat, positional but also has great attacking chances that lead to checkmate.
I'm looking for an opening to play against 1. E4, ideally with similar pawn structures and attacking chances. Any ideas?