So you want a new sink that can be selected for Games, which forwards to two other sinks.
Create a null sink:
pacmd load-module module-null-sink sink_name=games sink_properties=device.description=Games
and add two loopbacks from the monitor output of that sink to the sinks you want:
pacmd load-module module-loopback source="games.monitor" sink="name_of_built_in_stereo_sink"
pacmd load-module module-loopback source="games.monitor" sink="name_of_obs_sink
You can get the names with
pacmd list-sinks | grep name:
and omit the angular brackets.
Try on the commandline first, then add to Pulseaudio initialization files when it works the way you want.