Apple - How to Configure extra buttons on mouse
If you just want the side buttons to do back/forward in ANY app/environment, this app is the most basic, streamlined one I've found for this purpose on Mac OS X, and completely free (donate!). http://sensible-side-buttons.archagon.net/
This answer was copied from here: https://superuser.com/a/1230059/51328
I made Mac Mouse Fix.
It lets you remap your Side Buttons to Windows style Back and Forward (just like Sensible Side Buttons - in fact I copied their code) or to Switching between Spaces. It also lets you remap the Scroll Wheel Button to some useful functions like Mission Control and it features a refined Smooth-Scrolling algorithm, which I think strikes a great balance between fluidity and control.
Mac Mouse Fix is very light on system resources and it's a System Preferences Plugin so there's no status bar item. I tried to make it as clutter-free as possible.
On top of that, it's 100% free, it doesn't sell your data or anything, it's lightweight, easy-to-use, and fully open source.
You can download it here, or check out the GitHub repo here.
You have found an unfortunate fact of life living in the Mac world, a lot of hardware is Windows only. Yeah you can plug it in but without the correct drivers you will be lucky to get even minimal use out of the device.
If your SiliconWin mouse is not working out of the box with your Mac and there are not drivers provided by the manufacturer then there is little you can do in macOS to properly configure the device.
There is a "Universal USB device Driver" that has been available on the Mac for a long time, USB Overdrive.
Though I have never used it myself I have seen lots of discussions that point to that as a solution to exactly the problem you have. It's not free, a license will cost you $20. I imagine you could download it and give it a try and see if it allows you to do what you need to do. But at twenty bucks it costs as much (or more) as a cheap mouse so the economics of the situation may not be in your favor.