Dark standard model linked though Higgs

Yes. Actually, it is also possible to have such a Dark Standard Model that is linked to the Standard Model only through gravity.

Original theories for the nature of dark matter assumed all of the dark matter to be made of a single particle. WIMPs and Axions are the two prototype examples. Since these have not yet been found, it is perhaps no surprise that also more complex "Dark Sectors" are being looked into. A simple extension beyond a single dark matter particle for example includes a fifth force that lives in that dark sector. "Dark Photons" are one particular and currently rather popular model of that class. The problem with all these more complex models is though that the experimental constraints are minimal or non-existent, and one enters philosophical thin ice when developing such complex models that have no foundation in any experimental data.

Entirely dark galaxies are unlikely. The science of structure formation in combination with extensive gravitational lensing studies and analysis of the cosmic microwave background rule out the hypothesis that most of dark matter would be in clumps exceeding a few solar masses.

Concerning lower-mass clumps, there are also significant constraints, again from gravitational lensing. The corresponding observations typically are referred to "MACHO searches", for MAssive Compact Halo Objects. It is thus not yet ruled out completely that there could be dark stars or dark planets, but indeed it is rather unlikely, given the strong constraints from these searches.

To summarize, while it is allowed by current data that there could be dark aliens wandering around their planets, existing constraints make this quite unlikely.