As what does "very much" function in "to be very much"?
Solution 1:
It acts as an intensifier, to add emphasis to the statement that is being made. You can't use very on its own in sentences of this type ("We are very a family"); you have to say very much.
Solution 2:
It's an adverb modifying (emphasizing) the verb be. Cf. certainly, surely etc.
We (are) (very much) a family.
↓ ↓
verb ← adverb of emphasis
HTH.