if the game does not have controller support, you have to use xpadder/joy2key or equivalent, as the game itself was built to not use a controller.
of the games that DO have controller support, are seperated in two categories, more like 2 and a halfish:
games that use the old standard for controller layout, that was multiplatform, Direct Input (Dinput), its a very generic loadout that is flexible to getting mapped to based on how many inputs you have available on a controller. you typically will see dinput used in older games that predate xinputs creation. to convert dinput to xinput, software like x360ce(any dinput), DS4Windows(dualshock controller), Wiinupro(wii u pro controller)
Xinput, is a newer standard, where controllers are mapped specifically to microsofts Xbox 360 Controller. generally speaking, most modern games with controller support use xinput. to map Xinput to Dinput, you can use software like Xinput Plus
the half is some game devs allow both Xpinput and Dinput. For instance, fighting games on PC recognize both inputs typically. Emulator devs also typically allow all controllers by design and flexibility.
Steam's controller API has the Dinput/Xinput conversion built into it, as it allows remapping optionally, and automatically remaps xinput to dinput and vice versa if a game is launched through big picture mode.