We consider stationary and ergodic tessellations X = Ξ
n
n≥1 in R
d
, where X is observed in a bounded and convex sampling window W
p
⊂ R
d
. It is assumed that the cells Ξ
n
of X possess random inner structures, examples of which include point patterns, fibre systems, and tessellations. These inner cell structures are generated, both independently of each other and independently of the tessellation X, by generic stationary random sets that are related to a stationary random vector measure J
0 acting on R
d
. In particular, we study the asymptotic behaviour of a multivariate random functional, which is determined both by X and by the individual cell structures contained in W
p
, as W
p
↑ R
d
. It turns out that this functional provides an unbiased estimator for the intensity vector associated with J
0. Furthermore, under natural restrictions, strong laws of large numbers and a multivariate central limit theorem of the normalized functional are proven. Finally, we discuss in detail some numerical examples and applications, for which the inner structures of the cells of X are induced by iterated Poisson-type tessellations.