This is a really fun question! :D I don't know about the quantile part, I am sure that if you think about it a bit you'll come to that conclusion yourself regarding MANOVA and MLMVE.
To give you some pointers:
- Treat ANOVA as synonymous to regression (ANOVA is regression after all).
- Think of a MANOVA as a fully nested non-crossed two-level hierarchical mixed model, but then ignore the "lower" level groupings. What are you left with?
I haven't seen an exact proof myself, but yeah I am sure you can pull this off to some level. Sorry, I don't have the exact answer but try it, it is really thought provoking!
For a good targeted introduction to the subject try the following research report : Cross-classified and Multiple Membership Structures in Multilevel Models: An Introduction and Review (2006). I found the Behaviormetrika paper by Hwang and Takane An extended multivariate randon-effects growth curve model (2005) and the Metrika paper by Nummi and Mottonen On the analysis of multivariate growth curves (2000) also quite helpful. (I liked the Metrika et al. paper slightly more, but in case you don't have access to it, the Behaviormetrika is also quite good and is based in the same methodology)
Try to understand what Multilevel (or Hierarchical $^1$) Mixed Effects model does. MANOVA just comes out as a natural subcase of it by restricting some terms in the random effects covariance structure.
$^1$ those terms are not synonymous in general but for your MANOVA question I think they are; hierarchical models are multilevel models but not vice versa (multilevel models can be crossed).