There is a clear similarity between Stiefel-Whitney and Chern classes, if one replaces base field $\mathbb R$ with $\mathbb C$, coefficient ring $\mathbb Z/2$ with $\mathbb Z$ and scales the grading by a factor of $2$. For instance, both can be defined by the same axioms (functoriality, dimension, Whitney sum, value on the tautological bundle over $\mathbb P^1$)

Is there a deep reason behind this correspondence? The best explanation I have so far is the structure of classifying spaces. Being Grassmanians, they admit Schubert cell decomposition (which is essentially an algebraic fact). For cohomology of complex Grassmanian, differentials vanish for dimensionality reasons, and for real ones they vanish when reduced mod $2$.

There is a number of similar phenomena, for instance, $BO(1,\mathbb R) = K(\mathbb Z/2, 1)$ and $BU(1) = K(\mathbb Z, 2)$ which says that in both cases, topological line bundles are completely determined by their first characteristic class.

Also, I have been told that Thom polynomials for Thom-Boardman singularities of maps between real or complex manifolds have the same coefficients when expressed in $w_i$ and $c_i$. Can these facts be explained in a similar way?

There is a clear similarity between Stiefel-Whitney and Chern classes, if one replaces base field $\mathbb R$ with $\mathbb C$, coefficient ring $\mathbb Z/2$ with $\mathbb Z$ and scales the grading by a factor of $2$. For instance, both can be defined by the same axioms (functoriality, dimension, Whitney sum, value on the tautological bundle over $\mathbb P^1$).

Is there a deep reason behind this correspondence? The best explanation I have so far is the structure of classifying spaces. Being Grassmanians, they admit Schubert cell decomposition (which is essentially an algebraic fact). For cohomology of complex Grassmanians, differentials vanish for dimensionality reasons, and for real ones they vanish when reduced mod $2$.

There is a number of similar phenomena, for instance, $BO(1,\mathbb R) = K(\mathbb Z/2, 1)$ and $BU(1) = K(\mathbb Z, 2)$ which says that in both cases, topological line bundles are completely determined by their first characteristic class.

Also, I have been told that Thom polynomials for Thom-Boardman singularities of maps between real or complex manifolds have the same coefficients when expressed in $w_i$ and $c_i$. Can these facts be explained in a similar way?