Your Mad Catz Tournament Edition Arcade Stick for PS3 will work as a DirectInput device on your PC.
DirectInput was the standard for PC controllers before XInput made available for both the Xbox 360 and Windows XP with Service Pack 1 (SP1) and later, and Windows Vista. XInput is the new API where your controller is recognized as a Xbox 360 controller.
Most new games will expect a Xbox 360 controller. You can use “Xbox 360 Controller Emulator” that will allow your controller (gamepad, joystick, wheel, etc.) to function as an Xbox 360 controller.
Your Mad Catz Tournament Edition Arcade Stick for PS3 will be recognized as a PS3 controller on the PS4, so it won’t work. PS3 sticks by default are not compatible with PS4. The game developers need to enable it.
LabZero developed a custom driver that allowed any stick that reads as a generic USB gamepad or joystick would work with the PS4 version of its fighting game Skullgirls Encore. Capcom used that driver to make PS3 arcade sticks compatible with Ultra Street Fighter IV.
Street Fighter V for PS4 is not released yet so nothing confirmed. Capcom Still Evaluating Legacy Controller Support for Street Fighter V
Update: Capcom confirms Street Fighter V will support PS3 arcade sticks Street Fighter V Day One Update Adds Legacy Controller Support, Alex Coming in March