The answer to why different players have different styles is already in the question: because they are different. As in all aspects of our lives, in a game of chess we have options, and in the same way that different people approach their life in different ways, so do the chess players.
For most of the casual and even some tournament players, knowing the style of their opponent is useless: they play the game in the same way against all of their opponents. In higher levels however, knowing the style of the opponent may benefit you since you can change the nature of the position in your opponent's detriment: one usually does not want to play sharp and tactical games against a younger player or a player famous for their attacking abilities, and similarly one would not want to enter dull and even positions against a strong positional player. In this regard, I would not say that they are defined for the sake of analysis, but on the converse, analyzing a player's games will give you an idea of their style (and hence grant you an advantage).
Hence, style can mostly be characterized by personal preference. Knowing one's style can help you when choosing new openings and lines, ideally you would want them to suit your style. It can also help during a game, in case one detects that the best moves lead towards a position that does not suit one's style, it would be desirable to muddy the waters (if one tends more towards tactical games) or to exchange pieces or fix the pawn structure (if one tends more towards positional games).
Using these advantages to improve one's game would be to know our limitations and weaknesses, and work on them (like most of the top players do). However, this is a somewhat subjective answer, since other players would simply choose to enter tactical variations when faced against a quiet opening, or in the converse to play solid against wild openings. Even if I prefer the first option, the second is a completely viable strategy (as evidenced by some of the best players having an extremely limited repertoire that they know in great depth).
Disclaimer: for illustrative purposes, only two styles have been mentioned above, and a short comparison has been made. There are obviously many different styles, and this is in no way an exhaustive list.