Comparing different dark matter (DM) models, we explore the DM influence on black hole (BH) accretion disk physics, considering corotating and counterrotating thick accretion tori orbiting a central spinning BH. Our results identify accretion onto a central BH as a good indicator of DM presence, signaling possible DM tracers in accretion physics. We analyze accretion around a spinning BH immersed in perfect-fluid dark matter, cold dark matter and scalar field dark matter. Our investigation addresses observational evidence of distinctive DM effects on toroidal accretion disks and protojet configurations, proving that BH accretion tori immersed in DM can present characteristics, such as interdisk cusp or double tori, which have usually been considered as tracers for superspinars and naked singularity attractors. Therefore, in this context DM influence on the BH geometry could manifest as superspinar mimickers. DM also affects the central spinning attractor energetics associated with accretion physics, and its influence on accretion disks can be searched for in a variation of the central BH energetics as an increase of the mass accretion rates.