Forcing as a new chapter of Galois Theory?

Yes and No... There are strong parallels between forcing and symmetric extensions and field extensions and this way of thinking has been fruitful. However, like in the case of general ring extensions and group extensions and similar problems, this analogy is not perfect and pushing the similarity too far may actually obscure what is really going on.

That said, symmetric extensions do indeed show a great deal of similarity with Galois theory. Some work, notably that of Serge Grigorieff [Intermediate Submodels and Generic Extensions in Set Theory, Annals of Mathematics 101 (1975), 447–490] shows that there is indeed a way to understand intermediate forcing extensions in a manner extremely similar to the way understand field extensions through Galois theory. Some have even pushed this analogy so far as to study some problems roughly analogous to the Inverse Galois Problem in this context, for example [Groszek and Laver, Finite groups of OD-conjugates, Period. Math. Hungar. 18 (1987), 87–97].

There are some very algebraic ways of understanding forcing and symmetric models in a more global sense. For example, forcing extensions correspond in a well-understood way to the category of complete Boolean algebras under complete embeddings. Moreover, the automorphism groups of these algebras plays a crucial role in our understanding of the inner model structure of forcing extensions. An even farther reaching approach comes from transposing the sheaf constructions from topos theory into the set-theoretic context [Blass and Scedrov, Freyd's models for the independence of the axiom of choice, Mem. Amer. Math. Soc. 79 (1989), no. 404]. One could argue that this suggests a stronger analogy with topology rather than algebra, but there are plenty of very deep analogies between Galois theory and topology.

The above is not a complete survey of these types of connections, it is just to demonstrate that connections do exist and that they have been looked at and useful for a long time. Because of the relative sparsity of the literature, one could argue that these aspects are underdeveloped but that is hasty judgement. The truth is that there appear to be disappointingly few practical aspects to this kind of approach, perhaps because they are not relevant to most current questions in set theory or perhaps for deeper reasons. For example, the inner model structure of the first (and largely regarded as the simplest) forcing extension, namely the simple Cohen extension, is incredibly rich and complex [Abraham and Shore, The degrees of constructibility of Cohen reals, Proc. London Math. Soc. 53 (1986), 193–208] and there does not appear to be a reasonable higher-level approach that may help us sort through this quagmire in a similar way that Galois theory can help us sort through the complex structure of $\overline{\mathbb{Q}}$.

It seems to me that the groups involved in producing models without AC (whether as groups of permutations of atoms or as groups of automorphisms of complete Boolean algebras) and the forcing constructions themselves are two rather separate things. The most frequent use of forcing is to produce models of ZFC (not just ZF), and then groups are not involved. On the other hand, the Fraenkel-Mostowski-Specker method of permutation models for the negation of AC involves only permutation groups (and normal filters of subgroups), not forcing. Groups and forcing come together in Cohen's method of symmetric models. In the original (and still most common) presentation, one has a group acting on the forcing notion. But even here, the groups and the forcing are less entangled than they seem. Vopenka and Hajek showed (in their book "Theory of Semisets") how to get Cohen-style models by (1) starting with a permutation model, (2) forcing over it, and (3) passing to the pure (or well-founded) part. In this presentation, the groups (and filters of subgroups) are only in step (1), and the forcing is only in step (2).

It seems to me that, if one wants to analyze forcing from an algebraic point of view, one should begin with the simplest case, where no symmetries are involved. The algebraic side of this is the study of complete Boolean algebras. Afterward, one can embellish the picture by adding group actions, either acting on the Boolean algebras or producing permutation models over which to force. The Grigorieff paper that Francois cited is an excellent place to start.

Though the OP asked about forcing, let me also mention that the non-forcing context of permutation models might be a better place to look for Galois-like phenomena. To begin, note that a normal filter $\mathcal F$ of subgroups of a group $G$ makes $G$ a topological group, in which $\mathcal F$ generates the neighborhood filter at the identity. The notion of symmetry with respect to $\mathcal F$ that is used in defining the permutation model is just continuity of the standard action of $G$ on sets.

Just a minor remark about the possibility of classifications:

Galois theory can begin by examining finite-degree algebraic extensions whose Galois group is finite and well understood. Later you can start considering larger and larger groups, but these are less well understood than before.

With forcing you often begin with "locally uncountable" groups (read: uncountability in the ground model). We add to the mix the fact that there may be new automorphisms in the generic extension (and often there are), this is not a good idea.

There is, however, something to be said for sure. I recently talked about this with a friend who (much like me) spends most of his time in an inner model without choice. We agreed that there is probably something Galois-like to say on the stabilizer groups of every $P$-name. Whether or not it is deep, we haven't got a chance to investigate further.

Lastly, I would add that the analogy of field theory and forcing can only go so far. The algebraic closure is unique (assuming enough choice, at least) whereas generic extensions of the same forcing need not be isomorphic.