No, here "entropic" is as in the entropic force that returns a stretched rubber band to its unstretched condition, which (as it tends to be scrunched a bit) is at a higher entropy.

https://en.wikipedia.org/wiki/Rubber_band_experiment

"The stretching of the rubber band is an isobaric expansion (A → B) that increases the energy but reduces the entropy"

[apologies for any reversed signs below, I think I caught them all]

In Verlinde' entropic gravity, there is a gravitational interaction that "unstretches" the connection between a pair of masses. When they are closer together they are at higher entropy than when they are further apart. There is a sort of tension that drags separated objects together. In Carney et al's approach there is a "pressure mediated by a microscopic system which is driven towards extremization of its free energy", which means that when objects are far apart there is a lower entropy condition than when they are closer together, and this entropy arises from a gas with a pressure which is lower when objects are closer together than when objects are further apart. Pressure is just the inverse of tension, so at a high enough level, in both entropic gravity theories, you just have a universal law -- comparable to Newton's -- where objects are driven (whether "pulled" or "pushed") together by an entropic force.

This entropic force is not fundamental - it arises from the statistical behaviour of quantum (or otherwise microscopic) degrees of freedom in a holographic setting (i.e., with more dimensions than 3+1). It's a very string-theory idea.

The approach is very hard to make it work unless the entropic force is strictly radial, and so it's hard to see how General Relativity (in the regime where it has been very well tested) can emerge.