Question: I am interested whether a concept for the multivariate equivalent to quantiles exists. 
In the univariate case, a $p$-quantile can be computed via the inverse of the cumulative density function, however, this cannot easily be translated into the multivariate setting. 
Whereas in the univariate setting we are able to denote the $\alpha$ quantile $p\in\mathbb{R}$ such that $F(p)=\alpha$. Given we have a $N$-variate distribution, isn't it possible to define an $N-1$ dimensional subspace (for example an ellipse) if we are talking about bivariate distribution) such that a similar statement can be made?

Answer: 
Since the question was asked (and answered) a lot of interesting work has been done. In particular the concept "center-outward quantile function" was introduced as an extension of the quantile function to $\mathbb{R}^n$ and even some promienent matematishans such as Figalli recently worked on the topic. (See for example <https://arxiv.org/pdf/1805.04946.pdf> and references therein.)

While I am not familiar enough with this concept to go into details I believe that one of its most attractive features is the following: If $B\_1$ is the unitary ball centered in zero and $U$ is (some sort of) uniformly distributed random variable on $B\_1$ then $Q:B\_1\rightarrow \mathbb{R}^n$ is the center outward quantile function of $P$ if and only if $Q(U)\sim P$. Moreover it is possible to recover the quantile contours (isolines), obtained as the images under $Q$ of the nested hyperspheres $\{\partial B\_r\}\_{ 0<r<1}$.
i find it quite funny, but K-NN (k-nearest neighbors) and SOM (Kohonen's Self-organizing maps) and "vector quantization" are all about the same. They all give you "Multivariate Quantiles" (or "multidimensional quantiles").