Pairings in Identity-based encryption vs. Attribute-based encryption 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?
There really isn't a difference. It is just author preference in notation. Some authors prefer to write the pairing operations multiplicatively $e(P^a, Q^b)=e(P,Q)^{ab}$ while others prefer to write it additively $e(aP,bQ)=e(P,Q)^{ab}$.

This comes from the fact that in $e : \mathbb{G}_1\times \mathbb{G}_2\to\mathbb{G}_T$, $\mathbb{G}_1$ and $\mathbb{G}_2$ are (typically) elliptic curve groups, in which the group operation is additive, while $\mathbb{G}_T$ is a multiplicative group. Thus some authors might prefer to make the additive notation of the elliptic curve groups more explicit while other others prefer to make the notation more consistent.