Gamepad setup

I do not have a lot of experience with controllers or software and I am not exactly sure what is causing your issue, but I do use an old Playstation 2 controller with a USB adapter and the adapter messes up one of the sticks. I think it swaps left and right with up and down on one stick. I was able to fix it in a few games where I wanted to use the analog sticks by using x360ce. It stands for xbox 360 controller emulator. It is a free program that lets you use any controller and make the game think it is an xbox controller. It can be a little tricky to get it working, but it does allow you to set where are the keys are located on the controller. I am not sure if it will work with all games. I think the game has to support xbox controllers.

If I am trying to play a game that has no controller support, I will use the program xpadder and manually bind each key on the controller to the keyboard keys that are used in the game. Because the sticks will be bound to up, down, left, and right, I think this will limit the analog sticks to having 8 directions, the same as if you were using 4 directional buttons on the keyboard or the D-pad on a controller.

What controller do you have?

For Brothers - A Tale of Two Sons, it's not the game controller issue. The dev accidentally flip the controller movement. You can fix it

For Batman Arkham City I have the game set to "Disable Steam Input" (Properties->Controller->Dropdown setting "Disable Steam Input"). Some games native controller support doubled up with Steam Input mess up the controls, mainly stick sensitivity perhaps other things as well.

For that game I use a Xbone that is recognized as an Xbox 360 pad by the game.

Please try DS4 for windows.
Connect the controller, pop up the app and configure the buttons manually.

Maybe it'll help.

(allTrac's option is nice too, but ds4 is easier to get working with)

Xbox 360 Controller Emulator
https://www.x360ce.com/

This is an application that has been used to force some non-Steam DRM-free apps and games to have the gamepad complete.

Perhaps it is possible to configure these to work.
However, there is no way to guarantee the safety or behavior of these applications.