The Grassmannian $\mathrm{Gr}(k, n) = O(n) / O(k)\times O(n-k)$ describes all $k$-dimensional subspaces of $\mathbb R^n$. The product space $S=\mathrm{Gr}(k, n)\times \mathrm{Gr}(n-k, n)$ represents pairs of subspaces $(U, V)$.
Some of these pairs form a direct sum decomposition $U\oplus V=\mathbb R^n$, forming a subspace $$S' := \{(U, V) \in S\mid U\oplus V=\mathbb R^n \}.$$
I am quite sure that $S'$ is a known mathematical object, but I can't find any references on it. I checked N. Steenrod's Fiber Bundles, S. Watanabe's Algebraic Geometry and Statistical Learning Theory, T. tom Dieck's Representation theory and Googled "Grassmannian of pairs", "flag Grassmanian", "summand Grassmannian", but I haven't found anything.
Could you recommend me some references on this object?
As stated in the comment by @Deane you can easily describe the space of interest as the homogeneous space $GL(n,\mathbb R)/(GL(k,\mathbb R)\times GL(n-k,\mathbb R))$. The argument for this is that $GL(n,\mathbb R)$ acts transitively on the spaces pairs of subspaces $V,W\subset\mathbb R^n$ of the right dimensions such that $\mathbb R^n=V\oplus W$ and for the "standard pair" $V=\mathbb R^k$, $W=\mathbb R^{n-k}$, the stabilizer subgroup is the subgroup $GL(k,\mathbb R)\times GL(n-k,\mathbb R)$ of block diagonal matrices.
Howewer this does not correspond to the "orthogonal" interpretation of the Grassmannian as $O(n)/(O(k)\times O(n-k))$ that you start from. Indeed that space of decompositions $\mathbb R^n=V\oplus W$ is not a homogeneous space of $O(n)$ (since (roughly speaking)) $O(n)$ presereves the "angle" between $V$ and $W$. The corresponding homogeneous space of $O(n)$ is the space of orthogonal decompositions $\mathbb R^n=V\oplus_{\perp} W$, but this coincides with the Grassmannian itself, since any $k$-dimensional subspace determines the decomposition $\mathbb R^n=V\oplus V^\perp$ and this is the only orthogonal decomposition with first factor $V$.
So to get the Grassmannian nicely into the picture, you should view it as $GL(n,\mathbb R)/P$, where $P$ is the subgroup $\left\{\begin{pmatrix} A & B\\ 0 & C\end{pmatrix}:A\in GL(k,\mathbb R), C\in GL(n-k,\mathbb R)\right\}$. Then you can view $H:=GL(k,\mathbb R)\times GL(n-k,\mathbb R)$ naturally as subgroup of $P$ and correspondinly, you get a projection $G/H\to G/P$ (defined by $gH\mapsto gP$) which maps a decomposition to its first summand. Indeed, this is a fiber bundle with standard fiber $P/H$ (which naturally is an affine space).