I don't think I have anything really new to say here, but saying it again in different words might have some value, so I'll give it a try.
Consider these two seemingly-contradictory statements, both of which are known to be true:
An ordinary ferromagnet becomes spontaneously magnetized at sufficiently low temperatures, even though it does not have infinite volume.
On the other hand, a mathematical model of a something like a ferromagnet — the simplest example being the Ising model — does not show SSB except in the infintie-volume limit.
The resolution of this paradox is that a real ferromagnet does not exist in isolation. Even if we put it in a vacuum chamber, it still has a magnetic field, and the stuff outside the vacuum chamber (and in the walls of the chamber) can still be influenced by that magnetic field. Also, in order for the ferromagnet to cool down from above the SSB transition temperature to below that temperature, it must release energy into the environment. In the real world, there is no such thing as an isolated system. (Well, except maybe the whole universe, whatever that means.)
An accurate model of an isolated finite-volume ferromagnet predicts that SSB cannot occur, because the ground state is something like
$$
|\text{all spins up}\rangle +|\text{all spins down}\rangle
$$
rather than either term individually.
But in the real world, because the ferromagnet isn't isolated, the state ends up being something like this:
$$
|\text{all spins up}\rangle\otimes |E_\text{up}\rangle
+|\text{all spins down}\rangle\otimes |E_\text{down}\rangle
$$
where $E_\text{up/down}$ represent different states of the
ambient system which are almost exactly mutually orthogonal to each other.
Most importantly, their mutual orthogonality
cannot be undone by the action of any local operator
of manageable complexity. (Quantifying "complexity" here involves an unanswered question, but the qualitative idea is good enough here.)
This is decoherence.
When we consider a model of something like a ferromagnet (or other system exhibiting SSB) in isolation, we need the infinite-volume limit in order to get a mathematically strict version of SSB. But even in this case, there is a sense in which decoherence is still relevant. In effect, we're using the far-away parts of the ferromagnet itself as the "environment" with which the local part becomes entangled.
In the real world, the tunneling effects that would prevent a mathematically-strict version of SSB in finite volume become ineffective FAPP (for all practical purposes) when the volume is large enough, for the same reason that we can't observe superpositions of different measurement outcomes. In the wake of a high-quality measurement, the thing being measured has influenced its surroundings in a prolific way that we have no hope of ever reversing in practice. So... (wave hands here)... the superposition might as well have "collapsed" to just one of those terms. That's where I'll end this paragraph, because continuing any further down that path would take us straight into the philosophical quagmire through which no mortal physicist has ever successfully passed.
By the way, this is closely related to the motivation for using the cluster property to distinguish between the "real" SSB vacuum states and arbitrary superpositions of them. A vacuum state isn't just a lowest-energy state; it's a lowest-energy state that satisfies the cluster property. Arbitrary superpositions still have zero energy, but they don't satisfy the cluster property. Here are a couple of references that explain how the cluster property selects the "correct" vacuum states:
In the context of spin systems (like the Ising model): Section 23.3, "Order Parameter and Cluster Properties", of Zinn-Justin's book Quantum Field Theory and Critical Phenomena.
In the context of QFT: Section 19.1 in Weinberg, The Quantum Theory of Fields, Volume II.
