Higher-order exchange interactions and quantum effects are widely known to play an important role in describing the properties of low-dimensional magnetic compounds. Here, the recently discovered 2D van der Waals (vdW) CrI3 is identified as a quantum non-Heisenberg material with properties far beyond an Ising magnet as initially assumed. It is found that biquadratic exchange interactions are essential to quantitatively describe the magnetism of CrI3 but quantum rescaling corrections are required to reproduce its thermal properties. The quantum nature of the heat bath represented by discrete electron-spin and phonon-spin scattering processes induces the formation of spin fluctuations in the low-temperature regime. These fluctuations induce the formation of metastable magnetic domains evolving into a single macroscopic magnetization or even a monodomain over surface areas of a few micrometers. Such domains display hybrid characteristics of Néel and Bloch types with a narrow domain wall width in the range of 3-5 nm. Similar behavior is expected for the majority of 2D vdW magnets where higher-order exchange interactions are appreciable.