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Many models of dark matter have been proposed in attempt to ease the $S_8$ tension between weak lensing and CMB experiments. One such exciting possibility is cannibalistic dark matter (CanDM), which has exothermal number-changing interactions allowing it to stay warm far into its non-relativistic regime. Here we investigate the cosmological implications of CanDM and how it impacts CMB anisotropies and the matter power spectrum, by implementing the model within a linear Einstein-Boltzmann solver. We show that CanDM suppresses the small scale matter power spectrum in a way very similar to light Warm Dark Matter or Hot Dark Matter. However, unlike in those models, the suppression may happen while the CanDM model still remains compatible with CMB constraints. We put strong constraints on the interaction strength of CanDM as a function of its abundance for both constant and temperature-dependent thermally-averaged cross sections. We find that the CanDM model can easily solve the $S_8$ tension (but has no impact on the Hubble tension). Indeed, it can accommodate values of $S_8$ of the order of 0.76 while being compatible with CMB+BAO data. However, as long as the $S_8$ tension remains moderate, the overall $χ^2$ improvement is relatively small given the number of extra free parameters, and the CanDM model is not significantly preferred.