With over two dozen detections in the Milky Way, double neutron stars (DNSs) provide a unique window into massive binary evolution. We use the POSYDON binary population synthesis code to model DNS populations and compare them to the observed Galactic sample. By tracing their origins to underlying single and binary star physics, we place constraints on the detailed evolutionary stages leading to DNS formation. Our study reveals a bifurcation within the well-known common envelope (CE) formation channel for DNSs, which naturally explains an observed split in the orbital periods of the Galactic systems. The two subchannels are defined by whether the donor star has a helium core (Case B mass transfer) or a carbon-oxygen core (Case C) at the onset of the CE, with only the helium core systems eventually merging due to gravitational-wave-modulated orbital decay. We find that across different treatments of the CE phase, the formation of DNSs through both subchannels requires either a generous core definition of ≃30% H-fraction or a high CE ejection efficiency of α _CE ≳ 1.2. By testing different supernova kick velocity models, we find that galactic DNSs are best reproduced using a prescription that favors low velocity kicks (≲50 km s ⁻¹ ), in agreement with previous studies. Furthermore, our models indicate that merging DNSs are born from a stripped progenitor with a median pre-supernova envelope mass ∼0.2 M _⊙ . Our results highlight the value of detailed evolutionary models for improving our understanding of exotic binary star formation.