A 3-D storage media that can be read and written to might be referred to as a crystal by virtue of being a 3D grid of unit storage cells, no matter what the underlying technology may be.
It is more likely to be called a crystal if it looks like a gemstone: optically translucent. If it is read/written by lasers, that will be the case, in order to access the interior.
The storage cells would not be any kind of natural mineral. I can imagine a cell being a few atoms on each side of a cube, but carefully engineered to provide multiple stable states and mechanism to absorb the laser light and cause a change; something like a Chlorophyll absorbs light and drive an electron transport chain to eventually push the state into a different stable valley. But rather that being able to harness as much light as possible, it would be tuned to require precise frequencies simultaneously, allowing intersecting beams to address it.
(Actually, the intended cell size is too small to uniquely address that way. So a larger clump might have one "receiver" that takes coded data to a local grid of storage bits.)
The "state" can be stored in any number of ways that re-arrange a small number of atoms without adding or removing any: folding, flipping a group to a different permutation, orienting something differently within the surrounding structure, moving atoms to a different position. But state can also be stored as charge like in current flash memory or the perminant field in an "electoret" or magnet; or as energy levels within electron orbitals; or the quantum spin of electrons.