  It is commonly expected that for quantum chaotic many body systems the
statistical properties approach those of random matrices when increasing the
system size. We demonstrate for various kicked spin-$1/2$ chain models that the
average eigenstate entanglement indeed approaches the random matrix result.
However, the distribution of the eigenstate entanglement differs significantly.
While for autonomous systems such deviations are expected, they are surprising
for the more scrambling kicked systems. We attribute the origin of the
deviations to the local two-dimensional Hilbert spaces. This is also supported
by similar deviations occurring in a local random matrix model with global
diagonal coupling.
