It means it doesn't matter where it is: you can choose the origin, ie the point you measure from, it is arbitrary. Or another way of saying that is you can move the system to a different set of coordinates and it works in the same way.

... which means it's probably an imaginary physical system.

Maybe a good physical example is a piece of cloth that warps in 2D, and shrinks, when washed? Eigenvectors would describe the warping (skew, say) and eigenvalues the shrinkage relative to the original warp and weft.

Steve Brunton on YouTube has really good videos on eigenvectors & eigenvalues in context of matrix algebra (and then applied to simultaneous differential equations); https://youtube.com/watch?v=ZSGrJBS_qtc .