doesn't tem / há seis em nossa família make more sense? or is this just they way people would say this?
Not really, tem/há would need "pessoas" to complete the meaning (and it sounds a little remote); and yes, we do express ourselves like this.
Is "we are five in our family" a common usage in English? I wrote "we are five people in our family" because "we are five in our family" sounded weird to me. Maybe "we are a family of five" would fit better, even though it's not a literal translation.