I think nobody here will disagree to call Tal a master of attack and Petrosian a master of defense, or Kasparov more tactical and Karpov more positional. But then, such differences would be moot for a hypothetical 32-piece tablebase. Can, what we call "playing style" (cf. "Spielertypen" by Müller & Engel, claiming there exist 4 1/2 types) be formulated in objective terms? (Offhand example: make a statistic of how close their own officers are to the enemy king in an average game.) Links to scientific papers are welcome.
What is style?
Let's characterize it: Style is the preference for one move over an alternative objectively similarly (in the strict sense: equally) strong move, due to the nature of the resulting position.
EDIT: Note, that this formalized understanding of style is more abstract than just common high-level judgments, such as "X is a positional/tactical player". It does include such a statement, but also the 4-player-type standard model used by Müller&Engel, as well as e.g. certain decision patterns of engines that become important in the course of the answer. We need such a formalized understanding of the term style in order to substantially discuss it.
Can styles exist objectively?
The prerequisite for this is, that there exist positions, where there are more than one best (only considering best, because of the objectivity constraint) moves, i.e. moves that keep the current evaluation (win/draw/loss). If there is no choice, because there is only one best move, then there is no objective room for style. So, do such positions exist? Of course! Even in the endgame tablebases that we have solved so far, we have found, that there often are two or more distinct ways to e.g. convert a won endgame. If you agree, that the Open Games, the Sicilian, Caro-Kann, French (and most likely even second-grade openings such as the Pirc, Alekhine, Nimzowitch, Modern) all lead to a draw with perfect play, i.e. chess is a fair game and at least two from these openings cannot be refuted, then there is also the potential for style already from the very beginning of a game.
Do styles exist objectively?
This is the hardest question of the three, but with the help of what I wrote previously, I will attempt to give a reasonable answer. I agree with @Scounge's answer regarding the fact that "style" is hard to define by objective criteria and the lack of scientific studies. This is why my answer is more theoretical and formalized.
Preferences for certain types of positions and therefore style is subjective and interpreted by humans into objectively strong play. The famous "Boa constrictor" reportedly said about himself
Style? I have no style! (Anatoly Karpov)
Even though the vast majority of chess players saw a clear pattern (style) in his moves and dubbed him the "boa constrictor". Probably he wanted to say, that he plays objectively. Karpov also stated this, which falls exactly under what I consider style (in case both continuations were equally strong, see the first section of my answer):
Let us say the game may be continued in two ways: one of them is a beautiful tactical blow that gives rise to variations that don’t yield to precise calculation; the other is clear positional pressure that leads to an endgame with microscopic chances of victory. I would choose the latter without thinking twice. (Anatoly Karpov)
Engines come very close to the objective truth and are also considered to have styles and sometimes choose different moves (for example, preference for moving h-pawns, space or tactical play, etc.). The evaluation function is the main factor influencing style (either crafted by humans or self-trained with reinforcement learning) and determines the preference of engines for certain positions. Some positions are hard for engines to understand (i.e. evaluate correctly, e.g. closed positions, dead draws in materially unbalanced positions, ...) and they will have a preference for going into a draw they can easily calculate instead of going into a position where they are material down but can build a fortress, even though both are objectively equivalent.
In this position, Stockfish 14+NNUE has a very clear preference for Bxc2+, thinking everything else was very bad for Black, even though any legal move by Black leads to a forced draw (and there are many such positions).
[FEN "1k6/8/4p3/p1p3p1/P1P1P1P1/1K6/2B5/1b6 b - - 0 1"]
Openings have a huge impact on the nature of the position and the course of the game. Given that there are more than 2 unrefuted openings for a colour, the preferred choice of opening over alternatives can be considered a style choice.
Style is defined by the preference for some positions over other positions by chess players. This preference manifests itself in frequent decisions that lead to corresponding positions. There is proof in endgame tablebases and strong evidence in openings that styles can exist. It is your subjective discretion to judge whether a player's decisions are frequent enough to consider it his "style", i.e. whether it does exist for a certain player (who does not need to agree, see Karpov).
As the style is influenced by the capabilities (strengths/weaknesses) of the player (e.g. the understanding of fortresses by engines) and SF 14 would always choose one objectively best move over the other in the given example class (constructive proof), I come to the conclusion that styles do exist and will continue to exist forever, as chess will most likely never be solved.
Even if chess was solved and all players had perfect playing ability, they could have a preference for the quickest draw vs. the "most beautiful" draw (according to their subjective beauty standards, e.g. being down a lot of material). But in that case, chess was dead anyhow (no point in playing) and the question of style would become irrelevant. If only some players were able to always find the objectively best moves, then their style would become apparent also in the exploitation of the opponent's mistakes.
Therefore, with the given constructive proof for SF14 (which will never disappear as a chess player):
As long as chess exists as a game, there exist play styles.
I seriously doubt there are scientific papers on this. "Player styles" is probably impossible to define "objectively"/scientifically to the point where one can give a list of definite, rigid criteria that can be used by a computer to determine a player's style based on their games in a database such that everyone agrees that the computer is correct. In the end, what constitutes a distinct player style has to be arbitrarily determined at some point. In fact, I'd go so far as call an attempt to "scientifically define" player styles in this way a fool's errand.
The best kind of study one could hope for is one that tries to find clear statistically significant differences between groups of players (as chessplayers) who have different self-reported playing styles (out of a few listed), and in this sense of playing style I would be surprised if the study did not find clear differences between players who claim to be "attacking" vs. players who claim to be "solid", etc. How many broad "styles" that one should try to define is similar to how one should determine the average values of different pieces, i.e., one should do it by experience.
As a final note, one could also ask questions about objectively analyzing a specific player's unique chess style. As an example, if we feed every game that Magnus Carlsen has played up until 2021 into a neural network as training data, would the network be able to successfully pick out the games from TATA steel 2022 that were played by Carlsen? In other words, can a chess game bear the mark of Magnus Carlsen? The exciting thing about this question is that even if we are unable to describe in detail what "the mark of Magnus Carlsen" is, we can still test for whether it's there using this kind of method.
The only way to answer that question seems to be to perform a blind test. Can player A, who has played B many times before, recognize B (to whatever extent), in a series of blind games against him/her under similar playing conditions. Statisticians would have to decide just how many games that would be involve, and how many A-B pairs would need to play, as well as A-C pairs and games (where C is unknown to A, to guard against false positives, or decide just may be recognized) to produce a result with enough confidence.
If not, ... we probably are no further. But if there is a statistically significant 'recognition', we might as well call it 'style' as long as we don't have a better definition of the term.
I'm afraid that something like 1000 pairs of A-B people, 500 pairs of A-C people, and something like 50 games between each would be required, making a test impractical. But I'll leave that to the experts on statistics.
There are practical ways to measure complexity of positions using engine evaluation, such as for example depth of the search achieved in a certain time, or how much the evaluation changes when changing the depth. Statistics of complexity can be then used to measure style of players.
This has been studied in the context of evaluation of intrinsic strength of play for historic chess players, and it is exactly to account for players styles - a player who tends to create more complex positions more often also blunders more often, leading to lower "naive" accuracy scores. See, for example, this work (Jean-Marc Alliot, "Who's the master?") and the work it cites, in particular, Ken Regan's and Guid-Bratko.
As long as people play chess, there will be styles. Most people have a certain way to play, which is a style. It depends on what type of attack or defence you prefer, but if you make uncalculated moves, you will lose against a strong player.
Queen's side gives you a stronger position if played successfully. And then your pieces can eventually be homed in on the catsled king.
If you decide on the king's side, your style will be to attack faster and directly threaten the enemy king.
There are many substyles that you can choose from, but if you study them, there is a great chance you will win the opening and the game.