You will need another Chromecast device attached to the TV in your lounge.
But, yes, you can have MANY Chromecast devices all hooked up and running. I have three; one on the Den TV, one on the Living Room TV, and one on the Play Room TV.
One comment, you can only cast to one Chromecast device at a time from each app so you can't play on BOTH the bedroom and lounge at the same time. But, you can cast to the bedroom while someone else in your house casts something different to the lounge.