Yes, it’s quite common. Just Google “Congress and the Senate” (using quotes) and tons of examples pop up:
“In Congress and the Senate, some names have changed, but the adversarial math stays much the same.” (CBC)
“With Congress and the Senate having such a similar level of importance...” (comment on a Guardian article)
“Is congress more powerful than the Senate?” (question asked on WikiAnswers)
I don’t think you’ll see it so often from journalists—the CBC quote above appears to be an exception—as they have style guides that don’t allow it.