The naturality condition 1'' reduces the question of whether $v_1=w_1$ to the special case of the tautological bundles $\gamma_n$. In this case both $v_1$ and $w_1$ lie in $H^1(BO(n);{\mathbb Z}_2)$. This group is just ${\mathbb Z}_2$, as shown in the book of Milnor and Stasheff for example, or just from the fact that $\pi_1BO(n)=\pi_0O(n)={\mathbb Z}_2$.  So if $v_1(\gamma_n)$ is nonzero it must equal the nonzero class $w_1(\gamma_n)$.

This argument does not work for $w_2$ since $H^2(BO(n);{\mathbb Z}_2)= {\mathbb Z}_2 \times {\mathbb Z}_2 $ with basis $w_2$ and $w_1^2$, assuming $n>1$. Thus to characterize $w_2$ one needs to know more than just that $w_2(\gamma_n)$ is nonzero since $w_1^2$ also has this property, as does $w_1^2+w_2$. On the other hand, if one considers only oriented vector bundles then $H^2(BSO(n);{\mathbb Z}_2)={\mathbb Z}_2$ when $n>2$ so in this situation $w_2$ is characterized by naturality and being nonzero for the tautological bundle over $BSO(n)$.