As the global population grows and climate change intensifies, sustainable food production is critical. Marine aquaculture offers a viable solution, providing a sustainable protein source. However, the industry's expansion requires novel technologies for remote management and autonomous operations. Digital twin technology can advance the aquaculture industry, but its adoption has been limited. Fish net cages, which are flexible floating structures, are critical yet vulnerable components of aquaculture farms. Exposed to harsh and dynamic marine environments, the cages experience significant loads and risk damage, leading to fish escapes, environmental impacts, and financial losses. We propose a multifidelity surrogate modeling framework for integration into a digital twin for real-time monitoring of aquaculture net cage structural dynamics under stochastic marine conditions. Central to this framework is the nonlinear autoregressive Gaussian process method, which learns complex, nonlinear cross-correlations between models of varying fidelity. It combines low-fidelity simulation data with a small set of high-fidelity field sensor measurements, which offer the real dynamics but are costly and spatially sparse. Validated at the SINTEF ACE fish farm in Norway, our digital twin receives online metocean data and accurately predicts net cage displacements and mooring line loads, aligning closely with field measurements. The proposed framework is beneficial where application-specific data are scarce, offering rapid predictions and real-time system representation. The developed digital twin prevents potential damages by assessing structural integrity and facilitates remote operations with unmanned underwater vehicles. Our work also compares GP and GCNs for predicting net cage deformation, highlighting the latter's effectiveness in complex structural applications.