The external hydraulic pressure and internal medium pressure acting on submarine pipelines can lead to the coupling effect of active and passive constraints on the mechanical performance of steel–concrete double-skin composite tubes, resulting in a significantly different bearing capacity mechanism compared to terrestrial engineering. In this paper, the full-range concentric compressive mechanism of new-type stainless steel–concrete double-skin (SSCDS) composite tubes subjected to dual hydraulic pressure was analyzed by the finite element method. The influence of geometric–physical parameters at various water depths was discussed. The key results reveal that imposing dual hydraulic pressures significantly improves the confinement of double-skin tubes to encased concrete, resulting in a higher axial compressive strength and a non-uniform stress distribution; increasing the material strengths of concrete, outer tubes and inner tubes results in an approximately linear enhancement in axial bearing capacity; enhancing the diameter-to-thickness ratios of outer tubes and inner tubes can decrease the bearing capacity of SSCDS composite tubes; and the axial compression strength of SSCDS composite tubes with a higher hollow ratio of 0.849 tends to decrease with increasing outer hydraulic pressure. A practical method that integrates the effects of dual hydraulic pressures was developed and validated for the strength calculation of SSCDS composite tubes. This research provides fundamental guidelines for the application of pipe-in-pipe structures in deep-sea engineering.