The concept of 'torsion' pervades mathematics. As far as I know the origin of the word is in algebraic topology where it was used to describe chains $\gamma$ which are not boundaries but such that $2\gamma$ are boundaries. Then there's torsion in general abelian groups, rings, and modules. There's torsion in differential geometry, and analytic torsion. Lastly, there's $\mathrm{Tor}$, the left derived functor of the tensor product which is defined at least in the case of modules.
The lower dimensional $\mathrm{Tor}$ functors tell us about torsion. I don't understand what the higher ones do, but this bridge does exist. So the tensor product over of modules does poop out torsion from high above.
In differential geometry, the torsion form is often identified with a section of $TM\otimes \Lambda ^2T^\ast M$, called the torsion tensor. So formally, the tensor product pops up here too. Unfortunately
The definition of analytic torsion is beyond me entirely.
To what extent can these concepts be unified, seen as special cases of each other, or obtained from abstract nonsense?
It turns out that there is a relationship between analytic torsion and torsion in (co)homology. The idea is that analytic torsion equals Reidemeister torsion by the Cheeger-Muller theorem, and Reidemeister torsion is equal to the alternating product of sizes of torsion subgroups of integer homology (modulo some normalizing factors called regulators).
This relationship between the analytic and the algebraic is quite surprising! Unfortunately, it is not always well publicized in the analytic torsion literature (in my opinion), but it has gotten a lot of attention in recent research in number theory. It is explained (and applied) in, for example, this paper of Bergeron and Venkatesh: arXiv link.
To address your complete question, I don't know what hope there is of a "unified" theory of torsion. For example, I don't think that the torsion of a connection in differential geometry has anything to do with torsion of abelian groups.