Do general automorphic factors arise in some canonical way?

Yes, your comparison with "homogenous" functions on the Riemann sphere (that is, complex projective one-space), is good, insofar as the objects are not really functions on the thing, but "global sections of a vector bundle". Automorphic forms on the upper half-plane $H$ are (mostly) not $\Gamma$-invariant, so do not "live" on the quotient space $\Gamma\backslash H$.

On another hand, one of the modernizations of the theory of automorphic forms converts them to genuinely left $\Gamma$-invariant functions on a group, such as $SL(2,R)$, acting transitively on $H$, for example: for $f(\gamma z)=j(gamma,z)f(g)$, then $F(g)=j(g,i)^{-1}f(gi)$ is the conversion. It is an exercise to check that $F$ is left $\Gamma$-invariant. The automorphy factor/cocycle is converted into right equivariance under the maximal compact subgroup $SO(2)$ fixing the point $i\in H$. (One warning: for "half-integral weight", the automorphy factor is such that the automorphic forms lift to _metaplectic_groups_, rather than $SL(2,R)$.)

The latter context correctly insinuates that, apart from normalization and fooling-around, the collection of possible automorphy factors is actually indexed by the (irreducible) representations of the isotropy subgroup $SO(2)$ of the base point $i\in H$ (and similarly for other groups and spaces). Since this group is a circle, its repns are indexed by integers $k$, with $\pmatrix{\cos\theta & \sin \theta \cr -\sin \theta & \cos \theta}\rightarrow e^{ik\theta}$. In particular, there are obvious relations among these. Note that $cz+d$ at $z=i$ and with $c,d$ the lower entries of such a matrix becomes $-i\sin\theta+\cos\theta=e^{-i\theta}$.

(The $|cz+d|$ arises by tweaking, using $\Im(gz)=\Im(z)/|cz+d|^2$.)

So, yes, it is certainly reasonable to think about automorphic forms as "sections of a line bundle". Yes, the expressions $(cz+d)^k$ are special and canonical. Yes, even in more general circumstances (Siegel modular forms, Hilbert modular forms, ...) the line-bundle viewpoint is helpful.