Multi-sensor fusion is an effective way to enhance the positioning performance of autonomous underwater vehicles (AUVs). However, underwater multi-sensor fusion faces challenges such as heterogeneous frequency and dynamic availability of sensors. Traditional filter-based algorithms suffer from low accuracy and robustness when sensors become unavailable. The factor graph optimization (FGO) can enable multi-sensor plug-and-play despite data frequency. Therefore, we present an FGO-based strapdown inertial navigation system (SINS) and long baseline location (LBL) system tightly coupled navigation system (FGO-ILNS). Sensors such as Doppler velocity log (DVL), magnetic compass pilot (MCP), pressure sensor (PS), and global navigation satellite system (GNSS) can be tightly coupled with FGO-ILNS to satisfy different navigation scenarios. In this system, we propose a floating LBL slant range difference factor model tightly coupled with IMU preintegration factor to achieve unification of global position above and below water. Furthermore, to address the issue of sensor measurements not being synchronized with the LBL during fusion, we employ forward-backward IMU preintegration to construct sensor factors such as GNSS and DVL. Moreover, we utilize the marginalization method to reduce the computational load of factor graph optimization. Simulation and public KAIST dataset experiments have verified that, compared to filter-based algorithms like the extended Kalman filter and federal Kalman filter, as well as the state-of-the-art optimization-based algorithm ORB-SLAM3, our proposed FGO-ILNS leads in accuracy and robustness.