It's the "first" instance that opens - on my machine it corresponds to the window order in the taskbar. The feature to configure the started instances with parameters is not yet merged into Godot 4 but I think for 3 there's an addon in the asset library to give your started instances some parameters to help you identify them.
If you just want to quickly see which window you have in the "remote" tree view, I'd suggest a quick and dirty autoload script like this:
extends Node
func _ready():
name = str(randi_range(111,999))
# Chill for a frame in case some other script sets the window title
await get_tree().process_frame
DisplayServer.window_set_title("Window: %s" % name)
Put it at the end of the autoload queue and it will create a top level node with a random-number as the name and also put that name in the window title. Note that this is for Godot 4.
Alternatively if you don't want to write the number into the window's title you can also get the name property of the autoload and write in in a label somewhere in your UI.