Soft, porous mechanical metamaterials with pattern transformations have important applications in soft robotics, as well as tunable sound-reducing materials. To design new materials of this type, it is important to be able to simulate them accurately and quickly, in order to obtain their effective mechanical properties. Since standard simulations (usually using finite elements) can impose high computational costs, a graph neural network (GNN) that serves as a surrogate is developed, predicting the homogenized stress P, stiffness D and pattern transformation, for a given geometry of the representative volume element (RVE) and given loading F. As a test system, we consider a rubber-like material with a square stacking of circular holes, showing complex mechanical behavior. To respect relevant symmetries in terms of proper in-/equivariances, we propose a new similarity-equivariant graph neural network (SimEGNN). Our starting point is $E(n)$-equivariant GNNs (EGNNs) [1], because they can take a representation of the geometry as an input, allow for large deformations, and they respect some of the relevant symmetries: translation, rotation and reflection in-/equivariance. The SimEGNN additionally respects scale in-/equivariance (making it similarity in-/equivariant), can predict higher-order tensors (such as P and D), and respects periodic boundary conditions (such that the RVE can be shifted or multiple RVEs can be merged without changing the result). We show that incorporating symmetries of the similarity group yields significantly more accurate predictions and needs less training data compared to networks with fewer symmetries. [1] Satorras, V. G., Hoogeboom, E., and Welling, M. E (n) equivariant graph neural networks. International conference on machine learning. PMLR (2021, July) (pp. 9323-9332).