Two Uses for A Method of Interaction
Navigating the Interface
The mouse-style cursor on home consoles was often seen first as a result computer games that would have had natural mouse input being ported for play on a television screen with a controller. SimCity on the Super Nintendo and Command & Conquer on the PlayStation and Saturn recreated their original modes of input for lack of an alternative navigation scheme. In these cases, the "cursor on a console" was used not only for navigating menus but also for interacting with the game world. Genres that often used this method included real-time and turn-based strategy games, adventure games, and sim games. Because they were often used on game consoles that had digital directional input, the interaction felt imprecise and ill-suited for the software.
As the primary mode of menu navigation for modern games like Destiny or Gran Turismo Sport, the cursor on an analog controller emerges from higher fidelity analog interaction. Though some developers may use this style of mouse cursor to establish parity between computer and console versions of their games, it can typically be seen as a stylistic choice because it's usually more cumbersome than using buttons or a directional pad to move between interactable interface objects. From a hardware standpoint, the touch pad on the DualShock 4 and the Steam Controller afford the one-to-one mapping of a mouse cursor, though it is up to the developers of the game to take advantage of these features.
The cursor-style interface has been adapted to other, less precise uses as well. Both Pikmin and Halo Wars use a larger reticle for selecting things in the world because the player doesn't control a character that can directly manipulate objects.
Representing Computers in Game
Another way players will see a mouse cursor being represented in a console game is when there's a recreation of a computer environment within the game itself. If a player walks up to a computer terminal and interacts with it, they're taken to a diagetic interface that exists within the fictional world of the game. These games do not otherwise use a mouse cursor outside of this trope.