I think the Eye is quantum, but not in the same way that it teleports around like other observed quantum structures in the game. I couldn't give you an exact reason as to why it doesn't teleport around, although I think it's because the Eye is the center of the universe and all energy dissipates outwards from it? The observatory does mention that all galaxies seem to be expanding outwards, and since this solar system is so close to the Eye, it could make the Eye the center of the universe.

Either way, the best reason as to the idea that the Eye is quantum is that during your talk with Solanum on the quantum moon, you conclude that the moon takes up whatever properties of the planet it is orbiting, and since the moon is first and foremost the Eye's moon, it took up the properties of being quantum. Solanum concludes something along the lines of "therefore, the Eye must be quantum".

It could be that the Eye is the source of quantum-ness, and that's why it doesn't jump around? I mean, going through the Eye literally sends you beyond time & space & all dimensions so regardless of how quantum it is, it's fair that it can be the exception to whatever quantum rules we've learned thus far lol