If dark matter interacts with ordinary matter at all, it should most likely occur where ordinary matter is densest. Hence we have papers about neutron stars possibly containing dark matter cores (example).
But if neutron stars can have dark matter cores, white dwarfs, the Sun, or even the Earth can have dark matter cores too - they're just less likely to have such cores. If the Sun / Earth does have such a core dark matter would be much easier to study since they're so nearby. Is there any observational evidence that the Sun / Earth is 100% ordinary matter? If no, what is the observational limit of the Sun / Earth's dark matter fraction? I've seen popular-level articles (example) about such theories, but they're all rather speculative.
Answer
The easiest way for dark matter to become trapped inside another object is if it interacts and loses some kinetic energy. Otherwise it would just gain kinetic energy as it fell into a gravitational potential and then shoot out the other side. To be clear - this answer assumes that the "non-ordinary" dark matter that the question refers to is non-baryonic dark matter consisting of particles, as-yet unknown.
In order to interact we have to suppose some weak interaction of these particles is possible and if so this is going to be most effective when there is a large cross-section and interaction probability.
If the mass of an object is $M$ and its radius is $R$ and if the interaction cross-section is $\sigma$, then the following is illustrative.
The number of nucleons is $\sim M/m_u$. The number density of nucleons is $$n \sim \frac{3M}{4\pi R^3 m_u}.$$ The mean free path is $(n\sigma)^{-1}$ and the probability of interaction for a dark matter particle passing through the object will be $$ p \sim 1 - \exp(-2n\sigma R) \sim 2n\sigma R$$ $$ p \sim \frac{3 M\sigma}{2\pi R^2 m_u}$$
So for the Earth, putting in some estimates for $M$ and $R$, we have $p \sim 4 \times 10^{37} \sigma$; for the Sun we have $p \sim 10^{39} \sigma$; and for a neutron star with $M = 1.5M_{\odot}$ and $R=10$ km, we have $p \sim 10^{49}\sigma$.
Of course interaction alone is insufficient, the dark matter particle needs to lose energy and there are also considerations of gravitational focusing, the incoming energy spectrum (including the rest-mass of the particles) and the density of the dark matter and rate at which it might "build up". However, whatever $\sigma$ is (and we know it is small), it is 10 orders of magnitude more likely to interact and get captured inside a neutron star, than the Sun (or Earth). I suppose therefore the argument is that if there were any dark matter captured inside the Earth or Sun, then neutron stars must be full of the the stuff. It makes sense therefore to search for evidence of dark matter inside neutron stars.
Dark matter behaves gravitationally in the same way as ordinary matter, however it does not have the same equation of state as ordinary matter. There would therefore be structural differences (a different mass-radius relation) and also differences in the cooling rates for neutron stars (see this popular article for example). Given that we do not know what is at the core of a neutron star, then we don't know that there is no dark matter there. I will try to ascertain what observational limits do exist.
There have been various theoretical studies that show how the capture of dark matter might affect the structure of the Sun (e.g. Cumberbatch et al. 2010). Capture of dark matter lowers the core temperature and could have a potential effect on helioseismology results and the neutrino flux (especially at particular energies --Lopes & Silk 2010; Garani & Palomares-Ruiz 2017). No such effects have been unambiguously detected.
There also could be a neutrino signature from dark matter self-annihilation and this is a possible route to detecting dark matter trapped inside the Earth. Upper limits have been found from the ICECUBE experiment that are of course consistent with there being no dark matter there, but also consistent with the presence of dark matter with small self-interaction cross-sections (e.g. Kunnen 2015; Aartsen et al. 2017).
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