The bilinear map in Identity-based encryption should satisfy $e(aP,bQ)=e(P,Q)^{a\cdot b}$ whereas Attribute-based encryption schemes use $e(P^a,Q^a)=e(P,Q)^{a\cdot b}$ with $a,b\in\mathbb{Z}_p$ and $e:\mathbb{G}_1\times\mathbb{G}_2\rightarrow\mathbb{G}_T$.
Why are they different and are there reasons to prefer one over the other? Are they fixed in their purpose?