Percolation offers acknowledged models of random media when the relevant medium characteristics can be described as a binary feature. However, when considering habitat modeling in ecology, a natural constraint comes from nearest-neighbor correlations between the suitable/unsuitable states of the spatial units forming the habitat. Such constraints are also relevant in the physics of aggregation where underlying processes may lead to a form of correlated percolation. However, in ecology, the processes leading to habitat correlations are in general not known or very complex. As proposed by Hiebeler [Ecology {\bf 81}, 1629 (2000)], these correlations can be captured in a lattice model by an observable aggregation parameter $q$, supplementing the density $p$ of suitable sites. We investigate this model as an instance of correlated percolation. We analyze the phase diagram of the percolation transition and compute the cluster size distribution, the pair-connectedness function $C(r)$ and the correlation function $g(r)$. We find that while $g(r)$ displays a power-law decrease associated with long-range correlations in a wide domain of parameter values, critical properties are compatible with the universality class of uncorrelated percolation. We contrast the correlation structures obtained respectively for the correlated percolation model and for the Ising model, and show that the diversity of habitat configurations generated by the Hiebeler model is richer than the archetypal Ising model. We also find that emergent structural properties are peculiar to the implemented algorithm, leading to questioning the notion of a well-defined model of aggregated habitat. We conclude that the choice of model and algorithm have strong consequences on what insights ecological studies can get using such models of species habitat.