I have seen the notion of a delooping groupoid defined as a 1-groupoid having a single object * and the automorphisms of * to be the group G, with the composition of 1-morphisms being the usual group operation. In this sense, a group is a pointed 0-connected groupoid. In higher cases too, an n-group is a pointed connected n-groupoid.

But I have seen this for a discrete group G. What about a continuous group? Say, we take G to be U(1) and then form BU(1). Is it a 1-groupoid or a 2-groupoid? I have seen claims that BU(1) is a 2-group. I guess it follows from the fact that the homotopy type of U(1) is equivalent to B$\mathbb{Z}$, making BU(1) like B$^{2}\mathbb{Z}$. 

Could someone briefly explain how to see this as a 2-group? Do we have to always bring this to something relating to discrete groups?
Thanks.