I'm not sure if that's exactly what you mean by "being motivic", but this long exact sequence comes from a triangle in Voevodsky's category of (integral) motives $DM(\mathbb Q)$.
More generally if $X$ is a $\mathbb Q$-scheme of finite type, $Z\subset X$ a closed subscheme, $\pi : Y \to X$ the blowup at $Z$, and $E=\pi^{-1}(Z)$, then there is a triangle $$ M(E) \to M(Y)\oplus M(Z) \to M(X) \to M(E)[1] $$ in $DM(\mathbb Q)$. This is proved in Mazza–Voevodsky–Weibel's Lecture notes on motivic cohomology, see equation (14.5.3). It can also be viewed a formal consequence of the six-functor formalism for $DM(-)$, as described by Cisinski and Déglise in their paper Integral mixed motives in equal characteristic.