I initially wasn't even aware that this was an issue, but did notice that I found Sonic MUCH harder to control in SA2 than in the original Sonic Adventure and just assumed that this was just how the game played. After researching generic controller issues in the game I realised that although you can plug an XBox 360 controller straight in and play, unlike SA1 the analog stick functions much like a D-pad rather than a true analog stick, and that's why it seemed so hard to control Sonic and all the movements seem so erratic.

So the way to fix this issue is to enable XBox controller support explicitly in Steam. The approach to to do this is actually already described in the following guide: https://steamproxy.net/sharedfiles/filedetails/?id=1258435558
If you wish you can simply follow that guide to properly enable XBox controller support in the game and have the analog stick work as expected.

So why have I added this guide when the one linked to above already works? Because I almost didn't read it due to the title, as I didn't think I needed to "set up" a controller that already "worked" with Steam and was fine in other games like SA1.

It's worth stating that I actually only had to perform the following steps in order to get it to work, so you DON'T have to go through the whole guide that is linked to above:

1) Plug in your controller and then start Steam
2) Run the game at least once. In the configuration window for SA2 that appears choose the Player tab and select Gamepad instead of Keyboard
3) In the top right of Steam, go to Steam -> Settings. In the window that opens, navigate to the In-Game tab and ensure that the box next to Enable the Steam Overlay While In-Game is selected
4) Navigate to the Controller tab, and click on the General Controller Settings button
5) In the window that appears, check the box for XBox Configuration Support
6) That's it! You can check your controller setup for SA2 by navigating to your library, right-clicking SA2 and then hovering over Manage and choosing Controller Configuration.
7) Next time you run the game, just ensure that Gamepad is selected in the Player tab of the Configuration window that pops up as before.