On a dual-monitor setup, how can I keep the game in one monitor without the browser windows on the other monitor moving?

This is happening because the game you are running is not running at the same resolution as the monitor you are on. If you adjust the settings in the game to match those of the monitor, you shouldn't see this problem anymore.

Unfortunately, where it is not possible to match the resolution of your game to your monitor, you will still see this effect. Running it in windowed mode, where available will also bypass this problem, and many older games can be tweaked or modded to do one or the other.

The positions of the windows on the secondary monitor are based on the relative position of those window to the resolution.

Not using actual numbers here: So if you imagine that your monitors are each 1000 wide, the total width of both monitors is 2000. A window 1000 wide on the second monitor taking up the whole screen would occupy from the halfway mark to the edge. Now if you play a game full screen on the main monitor at a resolution of 500 wide, the total width is now only 1500. That window on the second monitor would still be positioned from the halfway mark to the edge. However the halfway mark is now at 750 and the window would also be 750 wide. If that window was forced to still have a 1000 width, then 250 of it would be off screen.

The solution to having things move around is to play games on the primary monitor at the native resolution of the primary monitor.