I recently read this post https://stats.stackexchange.com/questions/555855/why-is-a-regression-coefficient-covariance-variance, and I really liked @Marjolein Fokkema's intuitive explanation of Pearson correlation is. In particular, it was
The correlation coefficient tells us: If $x$ increases by $\sqrt{var(x)}$, how many $\sqrt{var(y)}$s will outcome $y$ increase? Thus, with a correlation coefficient of 1, an increase of 1 SD in $x$ is associated with an increase of 1 SD in $y$.
Now, I am wondering is there a similar explanation of covariance? Let's say that $Cov(X, Y) = C$, does that mean that an increase of $\sqrt{var(X)}$ in $X$ is associated with an increase of $C \cdot \sqrt{var(Y)}$ in $Y$ (in raw amount)?