Whether "Combinatorial Game Theory" (CGT) is considered a branch of "Game Theory" (GT) depends on who you ask/which book you check. But even though a few books on GT might claim that CGT is part of the umbrella of GT, or might rarely cover a strategy for a simple combinatorial game, the fact of the matter is that the material covered in the two fields are essentially completely separate.
For example, the GT texts "Games and Decisions: Introduction and Critical Survey" by Luce and Raiffa or "Introduction to the Theory of Games" by McKinsey contain essentially zero overlap with a text on CGT, and "Game Theory" by Maschler, Solan, and Zamir just has an early section about Chess and a version of Zermelo's Theorem, but no discussion of other combinatorial games.
To be glib, the connection between the two fields is almost entirely that they both have the phrase "Game Theory" in their name.
Sometimes GT is called "Economic Game Theory" as it is tied to some studies of economics. For example, the Stanford Encyclopedia of Philosophy says:
Game theory is the study of the ways in which interacting choices of economic agents produce outcomes with respect to the preferences (or utilities) of those agents, where the outcomes in question might have been intended by none of the agents.
Standard GT generally deals with situations where information is unknown, either because agents make decisions simultaneously, are privy to private information, have to deal with randomness, or some combination of things like that. In these situations, preferences beyond "winning is better than losing" become important.
A classic example of a simple game studied in GT is the Prisoner's Dilemma.
In contrast, CGT is almost entirely about two-player games of perfect information, alternating moves, and no chance. To quote "Combinatorial Game Theory" by Aaron N. Siegel:
The mathematical theory of combinatorial games pursues several interrelated objectives, including:
- exact solutions to particular games, usually in the form of an algebraic description of their outcomes;
- an understanding of the general combinatorial structure of games; and
- hardness results, suggesting that for certain games, or in certain situations, no concise solution exists.
A classic example of a simple game studied in CGT is Nim.