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.

Here's a good resource on the subject, from the lab of Casey Dunn, a siphonophore researcher.

Your argument is solid; in functional terms, the zooids in a colonial organism can be considered organs in an integrated body. The reason why we call these organisms "colonial" is actually a bit abstract: it has to do with the evolutionary/developmental history of these organisms.

You know the concept of homology? Like how your hand is homologous to a dolphin flipper, or the wing of a bat? All three organs can trace their history back to an original forelimb in the shared mammalian ancestor of all three animals. This is the key to understanding the definition of zooids, too. Each zooid in a colonial organism is homologous to the entire body of an individual in related, non-colonial organisms. For example, the reproductive zooid of a Portuguese man o' war is homologous to the entire body of a jellyfish.

It's a bit as if an offshoot population of humans evolved into grotesque creatures made up of hundreds of little human bodies, linked together, performing different tasks. Functionally, each of those little bodies can be viewed as just an organ — but looking at it from a perspective of homology, you've got a colonial organisms made out of human-shaped zooids.

(And, as noted in the linked text, all animals can be viewed as colonial in this sense, because each of our cells is homologous to the entire body of one of our single-celled ancestors.)

You can actually do this math for any species at all, without knowing anything about its biology except how many offspring they tend to have.

Think of it this way: animal and plant populations may fluctuate wildly from year to year, but in the medium-term, we can assume that the per-year average total population of a species is more or less stable — neither increasing or decreasing. This necessarily means that (assuming equal sex ratios, which most species have) each female is having, on average, two offspring per year that survive long enough to have their own offspring. That's one offspring per female to replace herself, and one to replace her male partner. Some females will have much more than two offspring survive to adulthood, and some will have none, but the per-female average is approximately two.

So if you find out that a species tends to have approximately 1000 offspring, you can divide 2 by 1000 to get the survival rate. If the typical litter size is four, you know that only half of them on average will make it. If a species tends to have a million offspring, it's two in a million.