Salt cavern underground gas storage (UGS) has attracted more and more attention worldwide for high peak shaving efficiency and high short-term throughput. To ensure the safe operation of this type of UGS, it is necessary to evaluate and analyze its stability. This paper investigates the influences of interlayers and cavern interactions on salt cavern UGS's stability. A 3D geomechanical model of double-salt cavern UGS with interlayers is established based on the geological data and creep constitutive relation of salt rock. Based on the long-term creep numerical simulation, the influences of interlayer number, interlayer thickness, interlayer dip angle, interlayer stiffness, cavern spacing, and cavern pressure difference on the deformation of caverns and stability performance of UGS are studied. The results show that the UGS with greater interlayer numbers has larger cavern deformation. The increase in interlayer thickness will improve the deformation resistance of caverns, but the effect is not obvious. The UGS with an interlayer dip angle of 12.5 deg has the best stability. Soft interlayer will decrease the deformation resistance of caverns, while hard interlayer has the opposite effect. In addition, the UGS stability can be enhanced by reducing the pressure difference between adjacent caverns. It is reasonable that the cavern spacing is twice the cavern diameter, which is beneficial to the UGS stability and will not cause a waste of salt rock resources. Finally, the corresponding production and construction control measures are discussed according to each factor's influence degree.