We analyze shot noise in a magnetic tunnel junction with a two-level quantum dot attached to the magnetic electrodes. The considerations are limited to the case when some transport channels are suppressed at low temperatures. Coupling of the two dot's levels to the electrodes are assumed to be generally different and also spin dependent. To calculate the shot noise we apply the approach based on the full counting statistics. The approach is used to account for experimental data obtained in magnetic tunnel junctions with organic barriers. The experimentally observed Fano factors correspond to the super-Poissonian statistics, and also depend on relative orientation of the electrodes' magnetic moments. We have also calculated the corresponding spin shot noise, which is associated with fluctuations of spin current.