Why to put variance around the mean line to the definition of $R^2$? By what is this particular choice dictated? Suppose we have a linear regression and we calculate $R^2 = 0.81$. That means $81\text %$ less variance around the regression line than mean line, since $R^2 = \frac{\mathrm{Var\ (mean\ line) - Var\ (regression\ line)}}{\mathrm{Var\ (mean\ line)}}$.
Question: why do we actually compare to the variance around exactly the mean line? Is it just the most natural choice, since we need to compare the variance around our fitted line to something? Why not to compare to the variance around something like $f(n) = 2n + 10$ instead (I've just invented it)?

The linear regression is predicting the conditional mean. Therefore, the natural choice for a naïve guess is just to predict the marginal/pooled mean of $y$, regardless of what the features are.
Therefore, the interpretation of $R^2$ involves how your model tightens up the guessing, compared to doing the most naïve sensible type of predicting.
As an analogy, if you had to predict the probability of a coin coming up heads, the naïve model would be to guess $0.5$. Then if you can examine the coin for weight distribution or if it has heads/tails on both sides, you can tighten up the guessing. Until you have access to that additional information, however, the only reasonable guess is $0.5$.
(I have heard that coins are not actually 50/50. Let’s ignore that for now.)
