Can Home Assistant tell which Google speaker a voice command came from?

Another way round it if you want to keep the Google speakers: put a presence sensor in each room (mmWave ones work best, motion sensors miss people sitting still) and have the command run a script that turns on the lights wherever someone is detected. It's not really knowing which speaker heard you, but for "turn on the light" the result is the same.

Not through the Google Assistant integration, no. As far as I know the command reaches Home Assistant without saying which speaker heard it, so "turn on the light" can't be mapped to the room you're standing in. The usual workarounds are saying the room out loud ("turn on the bedroom light"), or moving the voice side over to Home Assistant's own Assist with a voice satellite in each room. Those satellites are assigned to an area, so area-aware commands do work there.