Can a single player's choice of opening dictate a more (or less) aggressive game? Usually a player feels comfortable with few "positions" (unless he/she is a GM). That's why sometimes it's said that a player should choose and begin to study openings whose outcome results in a position he/she likes.
It's said, for example, that positional players should open with 1. d4, while aggressive players with 1. e4; a Black who chooses Sicilian 1. e4 c5 is more aggressive than one who chooses Caro-Kann, etc.


  Are there some lines which allow a single player to change the aggressive or positional path of a game, in order to surprise the opponent?


For example, a Benko Gambit, despite the "positional" 1. d4, is a double-edged opening, but it's chosen by Black whether to play it or not. Is there an aggressive or very tactical opening that White may choose after 1. d4?
Other examples: are there ways, for White, to lead a game to a strategic-positional slow battle when Black chooses a Sicilian Defense? Are there ways for Black to play positionally against a King's Gambit?
It feels as though you have already answered your own question. After 1. d4 d5, white may choose to offer the Queen's Gambit, and go for a somewhat more tactical variation, though black will likely decline. However, as you mentioned, white choosing 1. d4 is already a choice for a more positional play style. The idea is that after 1. d4 d5, you are already facing a closed game, which, in general, will lead to a positional game rather than a tactical one.

As for the Sicilian defense, white can choose to play a closed Sicilian by not offering up a pawn on d4. This is typically done with the move order 1. e4 c5 2. Nc3, intending to eventually gain some king-side space by not blocking the f pawn with the g1 knight. For this reason, the Sicilian defense is known as a Semi-open line.

Black has the option to decline the King's Gambit, though typically white will eventually (usually pretty soon) open up the position. Typically, the move order 1. e4 e5 leads to an open game, which in general is more tactical.

So, in attempt to answer the question: if you're playing white, and you're looking for a more tactical game, then you should probably open with 1. e4 rather than 1. d4. On the other hand, if you want a closed, positional game, I would suggest 1. d4. I hope this helps.