Is it possible to transfer information through strategic games? Let's say that there are 2 players Player A and Player B, engaged in a set of repeated strategic games. Is it then possible to transfer information from one player, say Player A, to the other player, Player B, simply by having Player B observe the strategies that Player A uses over a given period of time, in response to Player B's strategies? 

For example, Player B could form a simple hypothesis about player A's beliefs  and/or preferences, and test it out by using different strategies. The hypothesis can then turn out to be true or false (over a period of time, with some probability) if Player A's strategies in response to Player B's take on a certain character.

I would like to learn more about these kinds of games, if they've been studied.
What you are implying is thoroughly studied by what is called algorithmic game theory. There you can find the concept of a learning algorithm. In game theoretical scenarios where the mathematical equilibrium is difficult to evaluate, those algorithms are used to get an approximation.

What they do (in a simplified way)
1) Player A starts with an array of available strategies, each to be played with equal possibility.
2) In every iteration he randomly chooses one strategy and plays against his opponent.
3) The result comes as a feedback and according to an update function/algorithm, the array of strategies is re-evaluated/recalculated.
4) This is repeated many times, hopefully to converge to either one strategy which will be a nash equilibrium, or to a mixed-strategy profile.

It is a very interesting topic, connected a lot with machine learning and AI.
Here are some links to get an idea:

1) Game Theory &amp; Learning for wireless networks 

2) On Learning Algorithms for Nash Equilibria