Does the property of non-increasing slope be generalized to a concave function for multiple variables?

A common characterization requires you take two derivatives and work with the Hessian. I'm going to phrase this in the convex case (but the concave case is analogous, just multiply everything by $-1$).
By calculus, "$f^{\prime }$ is always increasing" is the same as saying "$f^{″}$ is always positive" (provided $f^{″}$ exists). You have observed that this property holds for convex functions. More generally, the following is a common result:
If a real-valued function defined on a real Hilbert space (e.g. ${\mathbb{R}}^N$) has a well-defined Hessian, $H$ then
$f\text{is convex}\iff H\text{is positive semidefinite}.$