If Steam is running in the background, it could be taking your controller input and remapping it to a keyboard and mouse input. You can quickly check by fully exiting Steam and seeing if your controller is detected correctly. If it is, you can try adding pcsx2 as a non-steam game like the other comment suggested (but then you would need to do this for every other non-steam game using a controller), or you could disable Steam Input for "Non-game controller layouts".

With your controller connected, open steam in the upper left, and open to Settings > Controller > Desktop Layout > Edit. From there, disable Steam Input, and see if PCSX2 starts detecting your controller as a controller instead of a mouse and keyboard.