One direct connection between the Euler-Mascheroni constant and the mean of the Gumbel distribution is the following:
Let $X_k \sim Exp(1/k)$ then the limiting distribution for the sum is for $n \to \infty$
$$-\log(n) + \sum_{k=1}^n X_k \to \text{Gumbel}(\mu =0,\beta = 1)$$
And from this point of view, as the limit for a sum of exponential distributions, the mean of the gumbel distribution can be expressed as the limit of the sum of the means of these exponential distributions, which is the limit of $H_n - \log(n) = \gamma$
$$\lim_{n \to \infty} E\left[ -\log(n) + \sum_{k=1}^n X_k \right] = \lim_{n \to \infty} -\log(n) + \sum_{k=1}^n \frac{1}{k} = \gamma$$
The coupon collector's problem illustrates intuitively this connection between the Gumbel distribution and a sum of variables. (Intuition about the coupon collector problem approaching a Gumbel distribution)
The view of extreme events as a sum could be seen by seeing the distribution of the extreme of a period of $k$ days and the extreme of a period of $k+1$ days as the latter being the former plus an extra little bit. In the case of exponential distributions you get that the mean of this extra little bit scales as $1/k$.