There's no single answer to this, it all depends on the game, the engine and the team(s) executing the engine and game.
Some typical ways to divide this though is
Some org has "engine" team and "game" team. Engine team implements features in the engine itself and game team implements game specific features using the embedded scripting language.
Engine features are game agnostic and game features are specific to games and are scripted / built-on top of the engine features.
But sometimes things get muddled as general purpose engines are used in games that require specific features in the engine itself. Typically the decision is then made by a) what kind of feature and how does it fit in the architecture of the engine b) what are the performance requirements c) who is willing to do the work and maintain it
Some typical ways to divide this though is
Some org has "engine" team and "game" team. Engine team implements features in the engine itself and game team implements game specific features using the embedded scripting language.
Engine features are game agnostic and game features are specific to games and are scripted / built-on top of the engine features.
But sometimes things get muddled as general purpose engines are used in games that require specific features in the engine itself. Typically the decision is then made by a) what kind of feature and how does it fit in the architecture of the engine b) what are the performance requirements c) who is willing to do the work and maintain it