There is a natural definition of a "connect-four monoid". I don’t know if it's mentioned in the literature, and I don’t know what it looks like, apart from some observations (below) that show that it is not a group. If there is a reasonably simple description of this monoid, it could lead to a “solution” of connect-four which is applicable to arbitrary board sizes. But it might be too complicated to be of any help.

To begin with, one should think about how the game naturally splits into components, and what it means for a game position to be the “sum” of its components.

As the game proceeds, the set of unoccupied sites can separate into components that do not interact with each other. In such a situation, a move consists in making a move in one of the components (this is like the classical theory), and winning in one component means winning the entire game (this differs from the so-called normal playing convention where the game ends when a player is unable to move).

In classical combinatorial game theory, games are classified into four outcome classes: Positive (Left wins no matter who starts), Negative (Right wins), Fuzzy (Player to move wins) and Zero (Player not to move wins). In a game like connect-four that can end in a draw, there are nine outcome classes, namely the functions from {Red to move, Blue to move} to {Red wins, Draw, Blue wins}.

Just as in the classical theory, the set of positions is an abelian monoid under addition (position means what the board looks like, without information about who is to move), and one may define two positions $X$ and $Y$ to be equivalent if for every $Z$, the games $X+Z$ and $Y+Z$ belong to the same outcome class.

For instance, all positions with an even number of unoccupied sites, and where none of the players can win, are equivalent. These might be called “zero-positions”, since they include the empty position (neutral element of the monoid).

The additive structure carries over to addition of equivalence classes, but unlike the classical case, it doesn’t become a group. The zero class is a neutral element, but not all positions have inverses. For instance, if $X$ is a position is such that the player to move (whether Red or Blue) can win in one move, then there can be no other position $X’$ so that $X+X’$ becomes zero, because no matter what we add to $X$, the resulting game will still be a win for the player to move.

On the other hand some positions clearly do have inverses. There is a class that contains all positions with an odd number of free sites, and where nobody can win. We might call this class $\star$, since it is similar to the class "star" in the classical theory. These positions are clearly not equivalent to the zero positions, since adding them will turn a “zugzwang” into a first player win. But we do have the equation $\star + \star = 0$.

There are several games, in particular misère games and card games, where the analysis of a corresponding monoid leads to a solution, see for instance my paper The strange algebra of combinatorial games and its references.

A couple of questions: Is the connect-four monoid infinite? What do the connect-two and connect-three monoids look like?

Finally, it seems worth taking a look at the Masters thesis of Victor Allis from 1988: A Knowledge-based Approach of Connect-Four.