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 (as algebraic varieties)][1]. Then a quadric surface can be regarded as a hyperplane section of the image $V \subset \mathbb{P}^9$ of the [Veronese embedding][2] $\mathbb{P}^3 \to \mathbb{P}^9$. Note that this is only *partially* analogous to the representation of a conic section (considered as a plane curve) as the intersection of cone and plane, since the conic section is related to the aforementioned intersection by a [projective transformation][3], which is a lot stronger than saying they are isomorphic as algebraic varieties.]


  [1]: https://en.m.wikipedia.org/wiki/Algebraic_variety#Isomorphism_of_algebraic_varieties
  [2]: https://en.m.wikipedia.org/wiki/Veronese_surface#Veronese_map
  [3]: https://en.wikipedia.org/wiki/Homography