In his paper Budgeting and Monitoring the Risk of Defined Benefit Pension Funds, Bill Sharpe writes:

[...] the sum of the weighted marginal risks of the portfolio components will equal twice the variance of the overall portfolio. This leads some to define the risk contribution of a component as half its marginal risk (that is, its covariance with the portfolio) so that a weighted average of these values will equal the variance of the overall portfolio. [...] it sometimes leads to an incorrect view that it is possible to decompose portfolio risk into a set of additive components and to incorrect statements of the form "this manager contributed 15% to the total risk of the portfolio".

In the next paragraph, Bill Sharpe continues to say that the additive decomposition does make sense if all the asset's returns were indepdendent.

What makes the view and the statements ((highlighted in italics) incorrect in general case, but correct when the asset returns are independent? It seems either way there is an additive decomposition of the variance, but not of the standard deviation, of portfolio returns.

$M_i = 2C_{ip}$ with $C_{ip}$ being the covariance between asset $i$ and the portfolio $p$. One can now argue that, since this covariance also depends on the other assets $j\neq i$, the value $M_i$ is implicitly connected to the other assets via the correlation structure.
On the other hand if all assets are uncorrelated, we have that $C_{ip}=C_{ii}w_i$, since $C_{ij}=0$ for $j\neq i$. Thus, the marginal contributions do not depend on the other assets. Therefore one can speak of a more "pure" contribution to variance in this case.