Two Controllers Plugged In, Why Does The Second One Show Up As Player One?

What you are seeing is normal, and a simple routine makes it predictable.

Browsers number controllers in the order they connect. The first one gets slot 0, the next slot 1, and so on, which is exactly what our tester shows. XInput, which most Windows games use for Xbox-style controllers, works in a similar way with four slots, and the controller's Xbox button light shows the player number it was given.

So the order changes because the connection order changes. A controller that wakes from sleep first, or a USB port that starts up first when the PC boots, takes player 1.

To lock it, make the order a habit. Start with both controllers off, turn on player 1's controller and wait until it is recognised, then turn on player 2's. If you play through Steam, its controller settings let you reorder controllers, and that order holds for the session.

On the one-controller game: that is the game. PC has no general one-gamepad limit; XInput supports four at once, and browsers and Steam support more. Some games simply support a single player only, or treat a second controller as another input for player 1. Check the game's store page for local co-op support before trying workarounds.