A unique crystal?

From the information given the polyhedron hast the same structure as a cube, but the angles are not all 90°. Since the edges all have the same length, it follows that the faces are rhombic. Given the value of the area, the angles of the rhombi are $\alpha$ and $180 - \alpha$ . It turns out one can indeed assemble such a solid (and they are actually present in some natural crystals):

You can either assemble 3 rhombi together at a vertex with their acute angle and the other 3 likewise and then paste them together (as in the image above). Or you can assemble the 3 rhombi together at a vertex with their obtuse angle and the other 3 likewise, and then paste them together.

Notice that there is a condition on the obtuse angle in order to assemble them at a vertex: the sum of the angles must be less than 360° (you cannot assemble 3 angles at a vertex in Euclidean space if their sum surpasses 360°). So if the 3 obtuse angles sum up to 360° or more, then there is only one possibility. This means $3 \cdot (180 - \alpha) \geq 360 \iff 180 - \alpha \geq 120 \iff 60 \geq \alpha$ So the largest angle $\alpha$ for which those conditions define a unique crystal is $\fbox{60°}$ .

Note: at $\alpha = 90°$ the polyhedron is again unique since then $180-\alpha = \alpha$ (it's a cube). But this value was excluded from the range.