Game theory, in its general and original sense, is a branch of pure and applied mathematics that studies the optimal or feasible strategies available to interacting players in various types of formalized decision-making scenarios (which often are, though they do not necessarily need to be, games).
Although a relatively young field of study (less than a century old), game theory has grown to encompass a broad range of subtopics, and has become an important tool (and a subject of study in its own right) within a number of sciences besides mathematics, including but not limited to economics, political science, psychology, biology and computer science. Within this broad field, a number of fairly distict subtopics may be recognized, such as:
Combinatorial game theory, which studies sequential games of perfect information, such as Chess, Go or Tic-Tac-Toe. These games are characterized by the property that they include no randomness or hidden information, and thus by the fact that, if all the players follow a perfectly optimal strategy (which must exist, in theory, even if it is not known), the outcome will always be the same.
"Classical" game theory, which assumes that all players are fully aware of each others' possible strategies and payoffs, but not necessarily of the specific moves others have chosen to make, and that they all choose their strategy independently to maximize their expected payoff.
Cooperative game theory, which allows players to collude and form (more or less) binding coalitions through some kind of enforcement mechanisms.
Evolutionary game theory, which replaces the assumption that players rationally choose their strategies with the assumption that they each follow a fixed hereditary strategy which they pass on, possibly with small mutations, to their offspring.
The point to note here is that all these subfields, and many more besides, fall under the broad umbrella term "game theory". In fact, pretty much any mathematical (or quasi-mathematical) analysis of games could be considered a part of game theory, except maybe for games of pure luck (which are traditionally considered to fall under probability theory, although some might see this distinction as more of a historical artifact than an actual qualitative difference).
As for the questions mentioned above, as a mathematician I would regard them all as falling under the scope of game theory, even if some of the answers here may not possess quite the level of mathematical rigor I would usually expect of a formal game-theoretical analysis.
Specifically, the following questions ask about optimal (or quasi-optimal) strategies of sequential games of perfect information, and are thus clearly within the scope of combinatorial game theory:
The remaining two questions may not be about combinatorial game theory, but do seem to fall under the scope of classical game theory:
The real problems with the game-theory tag here, IMO, are two-fold:
several of the questions are simply not very good questions, or, even if the questions might be potentially interesting, are poorly asked; and
even the game-theory questions that are reasonably well asked seem to rarely get good answers on this site, presumably because we don't have many mathematicians (or people from related fields) here to answer them.
Honestly, I suspect that several of these questions would've been better asked at math.SE, or possibly at MathOverflow. That said, I can also imagine valid game-theory questions that might get better answers here (e.g. because they required extensive knowledge of a particular board or card game that most mathematicians would be unlikely to have), so I'm not quite ready to say that the tag simply shouldn't exist here. But it is, IMO, sort of borderline.
Anyway, to summarize, I don't think there's anything wrong with the use of the game-theory tag on these questions. There might be other things wrong with them, but those issues should be resolved by editing or closing the questions, not by retagging.
Ps. We do also have several questions that are clearly about game theory in the mathematical sense, even if they're not (currently) tagged with game-theory. A few examples I found are Does a winning strategy exist for Don't Break the Ice? and
Are there guaranteed winning strategies for Quarto?.