The universal enveloping algebra of a Lie algebra $\mathfrak{g}$ is a flat deformation of $S(\mathfrak{g})$, so these algebras should be similar in many ways. Does at least this general similarity hold?

Yes, if a filtered ring $R$ has the property that its associated graded ring is Noetherian, then $R$ is Noetherian. Universal enveloping algebras have a PBW filtration such that the associated graded algebra is $S(\mathfrak{g})$. This is proved in Noncommutative Noetherian Rings by McConnell, Robson, Small  see sections 1.6 and 1.7. 

