While the behavior of the dominant component of the dark matter is reasonably well established by cosmological observables, its particle nature and interactions with the rest of the matter are not known. We consider three dark matter models that admit electromagnetic interaction between baryons and dark matter: (a) milli-charged particle (CCDM) of charge qccdm and mass mccdm, (b) a neutral atom of two charged particles of mass mdd (DD), and (c) a neutral atom of doubly charged particle and helium nucleus (HeD). We derive and discuss in detail the formation, stability, and interaction of these atoms with baryons. We derive the implications of this new interaction in the tight-coupling approximation, which allows us to analytically gauge their impact on the matter power spectrum and CMB anisotropy. We incorporate this new interaction into the publicly-available code CLASS to obtain numerical results. We compare our results with Planck 2018 data to constrain the fraction of interacting dark matter. For the range of allowed astrophysical parameters, the HeD atom yields the results of ΛCDM model for k < 1 Mpc−1, and hence its fraction is not constrained by CMB anisotropy data which is sensitive to k < 0.2 Mpc−1. For mdd ≳ 25 GeV, the DD atom is also not constrained by CMB data. For mdd = 10 GeV, the CMB data constrains the fraction of DD atoms to be smaller than 4% of the total CDM component. For qccdm = 10−6e and mccdm = 50 MeV, the CCDM fraction is constrained to be less than 1%.