The $GNOME_TERMINAL_SCREEN environment variable contains an object path for D-Bus.
It is used to address a tab in the Gnome Terminal when starting a process in it, and to be signalled of its termination.
You can see the relevant part of its D-Bus interface by running this command:
dbus-send --session --type=method_call --print-reply \
--dest=org.gnome.Terminal "$GNOME_TERMINAL_SCREEN" \
org.freedesktop.DBus.Introspectable.Introspect
The output (snipped for relevance):
[...]
<!DOCTYPE node PUBLIC "-//freedesktop//DTD D-BUS Object Introspection 1.0//EN"
"http://www.freedesktop.org/standards/dbus/1.0/introspect.dtd">
<!-- GDBus 2.64.6 -->
<node>
[...]
<interface name="org.gnome.Terminal.Terminal0">
<method name="Exec">
<arg type="a{sv}" name="options" direction="in"/>
<arg type="aay" name="arguments" direction="in"/>
</method>
<signal name="ChildExited">
<arg type="i" name="exit_code"/>
</signal>
</interface>
</node>
If you run dbus-monitor and open and close a Gnome Terminal tab, you can see the D-Bus communication in action.
The X Window System is not aware of what goes on in the D-Bus realm, and as far as I know Gnome Terminal does not expose any X specific information through D-Bus.
I have found one way to tie a process to the X window of a Gnome Terminal in which it is running, but it is less than ideal. Nonetheless, it may suffice for your purposes.
The idea as that when opening a Gnome Terminal window, we will generate an identifying value, and store it both in an X property of the Gnome Terminal window, and in an environment variable.
We can then later get the environment variable from the environment of the process (via /proc/<pid>/environ if needed), and scan the windows for the one which has our value in the X property.
As the window does not exist yet when opening a new Gnome Terminal, we can not set a property ourselves, but the gnome-terminal command accepts an option --role, and stores its value in the WM_WINDOW_ROLE X property of the Gnome Terminal window.
The purpose of the WM_WINDOW_ROLE X property is to uniquely identify windows belonging to the same client.
Without --role, Gnome Terminal assigns it a unique value, but you can do this yourself.
So here is a start-gnome-terminal wrapper, which you could invoke from the key binding which would normally start gnome-terminal:
#!/bin/sh
FINDWIN_ROLE=findwin-role-$(xxd -p -l 16 < /dev/urandom)
export FINDWIN_ROLE
exec gnome-terminal --role "$FINDWIN_ROLE" "$@"
To look through the windows for the property later, you could use wmctrl -l and xprop.
