The thing that makes quadric surfaces "3D analogs of conic sections" is just that they are defined by a single equation of degree 2. It's not a particularly helpful characterization though, I would say. It strikes me more as something a pedagogue would say in a (poor) attempt to relate a new concept to one already known.
[One could see quadric surfaces as "slices" of a certain geometric object, analogously to conic sections, but only if you are only interested in them up to isomorphism. Then a quadric surface can be regarded as a hyperplane section of the image $V \subset \mathbb{P}^9$ of the Veronese embedding $\mathbb{P}^3 \to \mathbb{P}^9$.]