Indeed the structure is similar for a very good reason: Both of these are particular examples of a beautiful concept in mathematics.
You have some underlying manifold $M$. In addition to the manifold you have a family of vector spaces parametrized by $M$. In other words over each point of $p \in M$ you have a vector space $V_p$ and as you move around in $M$ the vector spaces also change in a smooth way. Together with some consistency conditions the structure you get out of this is whats called a vector bundle.
Now once we have a vector bundle, we can consider a "vector field". This is simply an assignment of a particular element $v \in V_p$ to each point $p \in M$ where again the assignment is done in a smooth way. In the math language this is called a section of the vector bundle. Now since we want to do calculus, it's natural to ask how to differentiate sections. There is a clear problem with that however: differentiation instructs us to subtract values of functions at two nearby points of your domain, but in this situation the images of two nearby points are valued in different vector spaces, therefore subtraction makes no sense! This is where the connection comes in. It's a tool that allows one to connect nearby vector spaces and thus define an appropriate notion of a derivative (covariant derivative). That's really all a connection is: a rule that allows one to connect nearby vector spaces ("parallel transport"). If you think about it, just this alone will lead you to the right index structure $A_{i\,\,b}^{\,\,a}$ (locally it's an "$\text{End}(V)$-valued one form") along with the right transformation property ("gauge transformation"). Here $i$ is an index of your underlying manifold and $a,b$ are indices of your vector space $V$ .
Now given a vector bundle $V$ over $M$ with some specified connection $A$, I can ask what will happen if I use this connection to take some initial vector and move it around some small loop in the underlying manifold. That's essentially what the curvature of the connection $A$ is describing. In index notation it's written as $F_{ij\,\,\,\,b}^{\,\,\,a}$. One should also note here that curvature defined as such is always anti-symmetric in $i$ and $j$ ("$\text{End}(V)$-valued two form").
Now the specifics. In the case of Riemann tensor, your underlying manifold is some smooth Riemannian (or Lorentzian in the case of GR) manifold $M$ and the vector bundle is simply the tangent bundle (i.e the possible arrows/directions one can draw at a given point on $M$) and sections are simply vector fields on $M$ (think of a sphere with some arrows tangent to it at each point). Given this structure, there is a unique connection that makes sure that the smooth and Riemannian structure get along well, which is the well-known Levi-Civita connection (Christoffel symbols) and the curvature of this connection is the Riemann tensor.
In the Berry case, your underlying manifold is the space of control parameters that appear in the Hamiltonian and the vector bundle over the space of control parameters is simply the Hilbert space (or some subspace of "low energy states")/space of states at those values of parameters. The connection is given by Berry's connection and the Berry curvature is the curvature of this connection.
The reason you get some extra structure in the Riemannian case on the Riemann tensor is that the connection you define comes from (is uniquely defined by, in fact) an assumption known as "compatibility": the covariant derivative of the metric vanishes. A general connection on some vector bundle need not have such an extra structure, so you don't see this in the Berry case.
