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According to this preprint, over a connected proper algebraic variety $X$ there is a universal reductive group $G$ such that isomorphism classes of vector bundles of rank $n$ are in bijection with isomorphism classes of $n$-dimensional representations of $G$. Furthermore the component group is $\pi_1(X)$. So the cancellation property you seek holds for any ...
You mentioned the degeneracy subset definition. First lets re-establish the Grassman case as an example. Let $BU(n)=Gr_n(\mathbb{R}^{\infty})$ and take $e_i=(0,\dots 1, \dots )$ in the $i$-th position. Then define a vectorfield of the canonical bundle by $pr_{V}(e_1)$ at point $V$. Then the vanishing set is $Gr_n(\langle e_1^{\perp}\rangle)$. The fact that ...