Have you ever played dice with a friend? Betting a fixed amount of money each turn, you win if the number on the top face of the die is one, two or three, your friend wins in all the three other cases. At any single point in time, the sum of all the profits and losses amount up to exactly zero. This is an example of a very simple zero sum game.
A more precise definition is that a zero sum game is a [mathematical representation of a] situation in which a participant's gain (or loss) of utility is exactly balanced by the losses (or gains) of the utility of the other participant(s).
On a first observation, the combination of the example and the more formal definition leads to the conclusion that the utility is represented by the amount you win or lose. This is the first misconception. In many cases it is convenient to assume that this equality holds (like in the game of dice) but that is not generally the case.
Think of a large multinational institution that has exposure in many different countries, hence exposed to fluctuations in the respective currency rates. For proper financial planning this institution hedges a part of its exposure. The counter party might be a normal retail trader (although quite unlikely) with a different goal in mind. The utility of these two counterparties could not be more different: the institution 'does not mind' losing (regarding its utility) while the retail trader wants to profit from this single trade. In this scenario, the utilities of the two parties are not offset by each other because if the hedge is carried out perfectly, the institution is 'flat' while the retail trader wins.
Regarding utility, all trading activity is by definition a positive sum game: when two parties agree to make a trade, each of the parties considers the good it is receiving as more valuable than the good it is delivering.
For the further discussion, let's assume that utility is represented by the financial loss/profit. A short term trader can sell his position to any another participant, regardless of the investment horizon of the counterparty. Contrary to playing dice, there is no 'round end' when one sums up profits and losses. This means that the short term trader as well as the long term trader (as well as the broker/market maker) can make a profit on the position. The markets are an infinite (and not finite) multiplayer game.
Finally, it does not make sense to view the FX market isolated from the 'rest' of the economy. The currency market as a whole can appreciate or depreciate relative to other measures of value (commodities, resources, …), i.e. both EUR and USD can appreciate with regard to grain. Hence, the trader long EUR and the trader long USD both make an objective profit (with regard to grain).
Trading forex is a positive sum game because market participants trade with different subjective measures of 'value' called utility and may have different time horizons, in such a way that both parties can realize a profit. This, however, does not imply, that the majority of traders lose money in the markets; this is due to other reasons and worth another article.