Pose registration is critical in vision and robotics. This article focuses on the challenging task of initialization-free pose registration up to 7DoF for homogeneous and heterogeneous measurements. While recent learning-based methods show promise using differentiable solvers, they either rely on heuristically defined correspondences or require initialization. Phase correlation seeks solutions in the spectral domain and is correspondence-free and initialization-free. Following this, we propose a differentiable solver and combine it with simple feature extraction networks, namely DPCN++. It can perform registration for homo/hetero inputs and generalizes well on unseen objects. Specifically, the feature extraction networks first learn dense feature grids from a pair of homogeneous/heterogeneous measurements. These feature grids are then transformed into a translation and scale invariant spectrum representation based on Fourier transform and spherical radial aggregation, decoupling translation and scale from rotation. Next, the rotation, scale, and translation are independently and efficiently estimated in the spectrum step-by-step. The entire pipeline is differentiable and trained end-to-end. We evaluate DCPN++ on a wide range of tasks taking different input modalities, including 2D bird's-eye view images, 3D object and scene measurements, and medical images. Experimental results demonstrate that DCPN++ outperforms both classical and learning-based baselines, especially on partially observed and heterogeneous measurements.