Are flat and non-flat geometry a legit terminology in machine learning and statistics?
These are terminologies from Mathematics, they are valid in any field.
What is the mathematical definition?
In mathematics, a (Riemannian) manifold is said to be flat if its curvature is everywhere zero; otherwise non-flat. This is very different than the definition of flat in geometry that you have referenced. According to that definition, only points, lines, and hyper-planes are flat.
For example, set $\left\{(t,t):t\in(-1,1)\right\}$ is a 1D flat manifold in ${\Bbb R}^2$, set $\left\{(t,t^2):t\in(-1,1)\right\}$ is a 1D non-flat (positively curved) manifold in ${\Bbb R}^2$, and a hypersphere is an $n$D non-flat (positively curved) manifold in ${\Bbb R}^{n+1}$.
Accordingly, a cluster with a (non) flat shape corresponds to a (non) flat manifold.
Here are some examples from the document.
Points are concentrated around (A) two 1D non-flat manifolds (circles) which are non-convex, (B) two 1D non-flat manifolds (arcs) which are non-convex, (C) three 1D flat manifolds (segments) which are convex, (D) three 0D flat manifolds (centers as points) which are convex.