Intuitive explanation for Kin- and Group- selection
First of all, there is a very heated debate about this in the field of social evolution at present, and you aren't likely to get a conclusive answer. One theorist may give you one answer, but another will vehemently disagree. I'll start by logically answering your questions in reverse order!

Question 2: Can you please provide an intuitive explanation of why kin- and group-selection are the same thing?

The first formal 'proof' that they are the same came from a paper by Queller in 1992. Let me give you the gist of what he found without being ultra rigorous. I won't focus on relatedness, since that was answered nicely by @falsum, "Price's equation tells us that this happens when the genetic variance between-groups is higher than the genetic variance within-groups. This is equivalent to saying that altruists tend to interact with other altruists and, accordingly, that the coefficient of relatedness increases." 

Queller explored the change in mean additive genetic (breeding) value for a trait over a single generation using Price's equation 

$
\Delta \bar{G} = \mathrm{Cov}(W,G)
$

where $G$ is the additive genetic value, and $W$ is relative fitness. Now, the key question is, how should we define fitness? If we treat it as a random variable, we can do a linear regression of fitness on a set of explanatory variables. What should we choose for the explanatory variables? One natural choice might be to predict the fitness of a focal individual based on the phenotype of the focal individual itself, and the phenotype of the individuals the individual interacts with socially:

$
W_{i} = w_{0} + \beta_{direct} P_{i} + \beta_{social} P_{j}
$

where $w_{0}$ is baseline fitness, $P_{i}$ is the phenotype of the focal individual, and $P_{j}$ is the phenotype of the focal individual's social partner. However, another way to predict the fitness of a focal individual might be to focus on the average phenotype of individuals in the focal individual's group, and the deviation of the focal individual's phenotype from the mean phenotype of individuals in the group

$
W_{i} = w_{0} + \beta_{deviation} (\bar{P}_{group} - P_{i}) + \beta_{group} \bar{P}_{group}
$

These regression equations for fitness hold exactly (only for a single generation), independent of the 'true' form of the fitness functions (they can be as non-linear as you like). What these allow us to do, however, is partition fitness effects into those that are due to benefits of a behaviour, those that are due to costs of a behaviour (and unmodelled residuals or 'noise'). 

From a kin-selection perspective, it turns out that the benefit in Hamilton's rule is

$
\beta_{social} = B
$

and the cost in Hamilton's rule is

$
\beta_{direct} = C
$

From a group-selection perspective, it turns out that the benefits and costs in Hamilton's rule are distributed over deivation and group effects

$
\beta_{deviation} = C - B
$

and

$
\beta_{group} = C + (N-1)B
$

Thus, we find that it possible to write fitness equivalently in two different ways that relate the change in frequency of individuals to benefits and costs associated with a social phenotype. 

Question 1 Is group selection equivalent to kin selection for any evolutionary game or exclusively for the prisoner's dilemna?

They are always equivalent no matter what game is being played or how the population is structured if you are happy use the regression methods described to define fitness, benefits, costs, and relatedness. Many people have issues with the regression methods though. See Allen et al for a discussion of this. Others argue that kin-selection holds in every case and that the problems with kin-selection stem from a misunderstanding of the methodology. See Gardner et al for a discussion of this. 

Many biologists find the debates in social evolution about kin/multi-level selection unhelpful (at best), and choose either framework to work in based on the question at hand. A few are very partisan on either side. There is a great deal of misrepresentation floating around about both, so read papers making strong claims with skepticism and caution.