Yes, they're typically genetically identical. Typically a colonial organism begins its life as a single embryo, which is divided up as it develops, budding off new zooids that remain semi-attached (sharing body fluids, nutrients, etc.) but develop independently from that point on.

New zooids being created through budding is a homologous process to asexual reproduction through budding in non-colonial organisms. For example, the zooids in a coral colony are born through budding from an adjacent zooid; corals are related to sea anemones, many of which use the same process to reproduce. The main difference is simply that in sea anemones the new polyp breaks off and leads a completely independent life, whereas in corals the new polyp stays semi-attached and acts as a zooid.

This is somewhat analogous to how cells — both those in your body, and those in single-celled organisms like bacteria — reproduce by splitting apart, but cell division and whole-body budding are two separate processes in an evolutionary perspective.

Colonial organisms can often reproduce sexually, too, but at the whole-colony level, not at the zooid level. In other words, zooids within a colony do not have babies with one another that later join the same colony (there is no such thing as a zooid "joining" a colony, other than by budding); rather, reproductive zooids release sperm and eggs, which meet and become an embryo, and that embryo becomes an entirely new colony, as described above.