In a game where there is a scoring function, one can always rank players beyond the winner into 2nd, 3rd, etc. The real question is whether positions beyond first have any value. That question depends on many things, including player psychology, game design, and external incentives. Here are some examples of ways that this varies between games:
Example 1: Poker
Poker is arguably a game of pure external incentives. You play with money, for money. The obvious scoring function is how much money players have at the end of the game, using how long players were in the game as the tiebreaker for eliminated players. One of the significant variants of poker is how cash is dolled out in the end. In one variant, known as a "cash game", players can leave whenever they want, taking with them however much money they have at that point. In anther variant often referred to as "tournament style", players play an elimination game until only one or a few players remain and the total money put in is given exclusively to the last player or the last few players.
Which variant is used causes significant difference in game-play. In cash games, the focus is on having an overall profitable strategy. Meanwhile, in tournament style, it is much more important to accumulate a drastic chip advantage and eliminate players, and there is usually a 1-on-1 showdown at the end (called "heads up") which plays very differently than poker at a larger table. Alternatively, if there is a big enough prize for second or third in a tournament style game, playing very conservatively and just trying to survive can be a very good strategy, despite it not doing very well in a tournament style game that only gives a a prize for first. While the mechanics of the game are the same between these variants, the strategies differ wildly.
Example 2: Diplomacy
Like Poker, Diplomacy is a game with both an obvious scoring function and player elimination. Players control supply centers, each of which supports one military unit. A player wins by controlling 18 supply centers and a player with no supply centers is eliminated. Also of importance is that Diplomacy is a notoriously long game, often taking about 8 hours with the full 7 players if played to completion.
The rules of diplomacy describe the game as only one player wins, but in casual settings, players often form alliances and decide to declare their alliance as winning the game rather than playing the game out to its ultimate conclusion. In tournament Diplomacy, there is a provision for all players remaining in the game to declare joint victory, in which case all remaining players get a number of points equal to a fraction of the players remaining (so three players remaining in the game would each get 1/3 of a win). In a casual setting however, it is often hard for players to emotionally differentiate between 1/3 victory and 1 victory, especially given playing to full resolution can take another several hours. Whether or not players consider a joint victory to be a true victory causes massive differences in the integrity of alliances, since playing to true victory requires breaking alliances at some point. This causes huge differences in the outcomes of casual diplomacy games.
Another important factor in diplomacy is that if players have set aside a particular amount of time to play, getting eliminated early can be of significant dis-utility because then they are no longer part of the social event of the game. This can cause significant differences in game play, including players willing to make disadvantageous alliances simply to keep themselves alive. I played one game of diplomacy in which one player made a permanent alliance with me claiming he "just wanted to come in second"; it turned out to be true, and ended up wildly unbalancing the game in my favor.
Example 3: Archipelago
Archipelago is an unusual game in that it is semi-cooperative. In most cases, a game ends with all players winning or losing together, but then there is also a points system used to define the "grand winner" in the case that all players win. Having played Archipelago, I have difficulty emotionally differentiating between "losing" and "winning but not coming in first". The game explicitly defines a "second place" that most players get, but it still kind of feels like losing. I care much more how close I came (points-wise) to being the grand winner in Archipelago than about the fact that I nominally won since how many points I have is an indicator of my performance. I believe the game would be just as good if it was defined as "one player wins or everyone loses" instead of defining grades of victory.
Example 4: Games with Kingmaking
There is a fairly lengthy discussion on kingmaking here. The relevant aspect of kingmaking here is that players not in the running to win become rogue agents since they are no longer motivated by the stated goal of the game. This highlights that players usually are trying to achieve the stated goal of the game. Players then have secondary goals, which may include implicit standing at the end of the game, but can also include getting revenge on a player that screwed them, helping a person they like better, or doing something that amuses them. Kingmaking is one of the results of the enormous complexity underlying players' actions in games. The gist of it is that players' motivations are far more complex than a univariate utility function.
Asside: Tournament Match Fixing
There is lots of external discussion on the topic. The relevant aspect of match fixing here is that players can be willing to lose games given proper external incentives. Consequently, tournaments often have very severe penalties against it to preserve the intended incentives of the game.
Using Score for Future Games
The implicit places beyond first are a good approximation for the strength of your strategy, assuming that the outcome of a strategy is normally distributed. If you "come in last" in a game, it could be a result of bad luck or it could be a result of bad game-play, but either way usually requires a very different post-game analysis than if you "came in second". Coming in last often requires rethinking your entire approach, while coming in second may just be about tweaking the finer points. But even here, we care more about margin of victory than we do about ranks. If you "come in second" in a game of Catan, with the final scores being 10, 9, 6, and 3, you did much better than if you "come in second" with the final scores being 10, 5, 4, and 3.
Of course, this becomes less useful if the outcome of the game given a strategy is not normally distributed. Taking a single incredibly risky move that causes you to win a lot or lose a lot all at once (such as picking up an enormous pile of the discard in Rummy 500, keeping a very difficult set of routes in Ticket to Ride, or trying to shoot the moon in Hearts) will usually cause you to come in first or last depending on whether or not it succeeds. These moves are typically only good if coming in first (i.e. "winning) is all that matters, since if they fail, they will be very detrimental to your points. Taking these moves also makes your final points value a poor indicator of the performance of your strategy since the average is a worse approximation of the strategy's chances of success. For example, a strategy that scores 100 points 75% of the time and 0 points the remaining 25% will win far more games than a strategy that reliably scores 90 points, despite having a lower average score.
Players have very complex utility functions when playing games. Some players try to win, no matter the cost. Some players try to do as well as possible relative to other players, and care about "coming in second" even if the game doesn't say that it counts for anything. Some players care about doing well relative to their past performance so that they can improve at the game. Some players will use the scores of the players as a tool to analyze the strategies employed by each. Some players play for motivations unrelated to the stated purpose of the game, such as to do something amusing or to achieve some external social goal. There is lots of overlap between these categories, and lots of things (like prizes) that can change players' utility when playing a game. I believe that implicit ranking is real and is always a factor in playing a game, but how much of a factor is entirely situational.