Dataset to give same eigenvectors?
I will restrict myself to the finite-dimensional situation. 

First, if we are talking about normal eigenvalues and not generalized ones, we need a square matrix. If $M\in\mathbb{R}^{n\times n}$ (or $\mathbb{C}^{n\times n}$) is a diagonizable matrix, it has an eigendecomposition
\begin{equation}
M = V\Lambda V^{-1}
\textrm{,}
\end{equation}
with $V:=\begin{bmatrix}v_{1},v_{2},\ldots,v_{n}\end{bmatrix}$ the matrix with its $n$ eigenvectors $v_{i}$ as columns and $\Lambda :=\operatorname{diag}\left(\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\right)$ the diagonal matrix with the corresponding eigenvalues on its main diagonal.

As you can see from this equation if you want to get matrices with the same eigenvectors, i.e. $V$ fixed, you can only change the eigenvalues $\lambda_{i}$. In general the relationship between the eigenvalues and the entries of the original matrix $M$ is non-trivial. 

The easiest situation arises if the eigenvectors $v_{i}$ form the standard basis, i.e. only $1$ at the $i$th position, $0$ otherwise. Then $M=I\Lambda I^{-1}=\Lambda $ ($I$ the identity matrix) and you can change every entry on the main diagonal without changing any eigenvector. 

Edit

I understood the question differently than victor. I thought the question asks what kind of different datasets can be described by the same eigenvectors.