Arduino Bluetooth with game controller

You need a module that supports the Bluetooth HID profile. The HC-08 module appears to function as a UART designed for BLE applications. Unless there is a different firmware for it, you probably won't be able to use it as an HID.

Searching google for "bluetooth hc-08 hid" suggests that the hc-05 or hc-06 can be flashed with a firmware that supports the HID profile.