This article considers the fusion of target estimates stemming from multiple sensors, where the spatial extent of the targets is incorporated. The target estimates are represented as ellipses parameterized with center orientation and semi-axis lengths and width. Here, the fusion faces challenges such as ambiguous parameterization and an unclear meaning of the Euclidean distance between such estimates. We introduce a novel Bayesian framework for random ellipses based on the concept of a Minimum Mean Gaussian Wasserstein (MMGW) estimator. The MMGW estimate is optimal with respect to the Gaussian Wasserstein (GW) distance, which is a suitable distance metric for ellipses. We develop practical algorithms to approximate the MMGW estimate of the fusion result. The key idea is to approximate the GW distance with a modified version of the Square Root (SR) distance. By this means, optimal estimation and fusion can be performed based on the square root of the elliptic shape matrices. We analyze different implementations using, e.g., Monte Carlo methods, and evaluate them in simulated scenarios. An extensive comparison with state-of-the-art methods highlights the benefits of estimators tailored to the Gaussian Wasserstein distances.