Consider a determined linear system of differential order $k$ of the form $\sigma^{i_1\cdots i_k}_{ab}(x) \partial_{i_1} \cdots \partial_{i_k} u^b(x) + l.o.t = 0$. The coefficients $\sigma^{i_1\cdots i_k}_{ab}$, a square matrix in the $ab$ indices, constitute its principal symbol. By replacing $\partial_i$ by an indeterminate $p_i$ (like in the Fourier transform), the symbol defines a square matrix valued function on the cotangent space at $x$, $\sigma_{ab}(x,p) = \sigma^{i_1\cdots i_k}_{ab}(x) p_{i_1} \cdots p_{i_k}$, also referred to as the symbol. The locus $\mathcal{C}(x)$ of all $p$ for which the symbol fails to be an invertible matrix is the characteristic variety of the PDE at $x$.
Further classification is based on the geometry of $\mathcal{C}(x)$, which is usually assumed to behave somewhat uniformly with respect to $x$. If $\mathcal{C}(x) = \{0\}$ then the equation is called elliptic. If $\mathcal{C}(x)$ looks like a cone which has compact intersection with some affine codimension-1 hyperplane (a "spacelike" hyperplane) then the equation is considered in a rather weak sense to be hyperbolic (the geometric optics approximation could be used to construct waves traveling at finite speed). The simplest situation is when this intersection is smooth and is the boundary of some bounded convex set (e.g., a sphere). In general, the intersection can be multi-sheeted, non-convex, self-intersecting, etc. There is a gradation of notions of hyperbolicity corresponding to these variations.
In the usual case of a single scalar equation in two variables, the sign of the $A$-$B$-$C$-determinant can be used to determine the geometry of the characteristic variety. In principle, one could try to do something similar in higher dimensions, based on some more sophisticated invariants of the principal symbol coefficients $\sigma^{i_1\cdots i_k}_{ab}(x)$. But this will obviously become increasingly difficult in higher dimensions of dependent and independent variables.
Not much else can be said at this level of generality. But if you're interested in this level of generality, you'll find more information for instance in
Seiler, Werner M., Involution. The formal theory of differential equations and its applications in computer algebra, Algorithms and Computation in Mathematics 24. Berlin: Springer (ISBN 978-3-642-01286-0/hbk; 978-3-642-01287-7/ebook). xxii, 650 p. (2010). ZBL1205.35003.