The microscopic theory of the temperature-dependent magnetization μ (T) of ferromagnetic nanostructures with relatively weak magnetic anisotropy proposed in [1] leads to a generalized form of the Bloch law different from the one presently used for the analysis of experimental and numerical simulation data. It is demonstrated with a specific example that the latter is misleading, despite of its good performance as a data fitting formula. In particular, it largely overestimates magnetic softening in nanostrutures and the true softening is better captured by the proposed form of the modified Bloch law. Magnetization consists of two contributions: the Bloch term for the bulk material, and the size-dependent part including the effects of geometric shape and coupling to the surrounding medium. Such coupling can strongly modify the properties of a nanostructure through a mechanism mediated by collective excitations (magnons). On the one side, magnons are sensitive to boundary conditions and at the same time involve all the constituent magnetic moments of the crystal; on the other side, they dominate the low-energy part of the magnetic excitation spectrum. Respectively, it is shown that boundary conditions are a source of large variation of magnetic polarization. For instance, embedding a free unpolarized nanoparticle in a polarizable medium may induce a magnetic moment which can exceed even the bulk value, an effect strongly enhanced by shape anisotropy. For an asymmetric shape one can increase or even decrease the magnetization of a free-standing nanoparticle depending on the contact area, suggesting an interesting possibility of controlling magnetization. It turns out that in the case of non-uniform coupling suppression of spin fluctuations on a part of the surface can be prevailed by the enhanced fluctuations of the other spins of the sample.