IMO, the scenario is closer to your (a).  I'll sketch an explanation of the duality between $H^1(E,\mathbf{Z}_l)$ and the dual to the Tate module.  We have $H^1(E,\mathbf{Z}_l)=\text{Hom}(\pi_1(E),\mathbf{Z}_l)$,  

where that $\pi_1$ means etale fundamental group with base point the origin $O$ of $E$.  Thus the isomorphism we really want is between $\pi_1(E)\otimes\mathbf{Z}_l$ and $T_\ell(E)$.  

What is $\pi_1(E)$?  In the topology world, we'd consider the universal cover $f\colon E'\rightarrow E$ and take $\pi_1(E)$ to be its group of deck transformations.  Then $\pi_1(E)$ has an obvious action on $f^{-1}(O)$.  If $E$ is the complex manifold $\mathbf{C}/L$ for a lattice $L$, this is just the natural isomorphism $\pi_1(E)\cong L$.  

But in the algebraic geometry world, there is no universal cover in the category of varieties, so the notion of universal cover is replaced with the projective system $E_i\to E$ of etale covers of $E$.  Then $\pi_1(E)$ is the projective limit of the automorphism groups of $E_i$ over $E$.  

One nice thing about $E$ being an elliptic curve is that any etale cover $E'\rightarrow E$ must also be an elliptic curve (once you choose an origin on it, anyway);  if $E\rightarrow E'$ is the dual map then the composition $E\rightarrow E'\rightarrow E$ is multiplication by an integer.  So it's sufficient to only consider those covers of $E$ which are just multiplication by an integer.  Since it's $\pi_1(E)\otimes\mathbf{Z}_l$ we're interested in, it's enough to consider the isogenies of $E$ given by multiplication by $l^n$.

What are the deck transformations of the maps $l^n\colon E\rightarrow E$?  Up to an automorphism of $E$, they're simply translations by $l^n$-division points.  And now we see the relationship to the Tate module:  A compatible system of deck transformations of these covers is the exact same thing as a compatible system of $l^n$-division points.   Thus we get the desired isomorphism.  Naturally, it's Galois compatible! 

In the end, we see that torsion points were tucked away in the construction of the etale cohomology groups, so it wasn't exactly a coincidence.  Hope this helps.