Nakagawa and Schielzeth (2013) call intra-class correlation $R^2$ as($ICC$) "closely connected to $R^2$" and refer to their 2010 paper. From posting this question I already seen few mentions of $ICC$ treated as "variance explained" since it in fact is a part of total variance explained by all the random effects and residuals. So it seems $ICC$ is used as an analog to $R^2$ while one have to remember that both statistics have their flaws (e.g. here and here).
Nakagawa, S. & Schielzeth, H. (2010). Repeatability for Gaussian and non-Gaussian data: a practical guide for biologists. Biological Reviews, 85, 935–956.