Integrated Sensing and Communication (ISAC) has been identified as a key 6G application by ITU and 3GPP. Channel measurement and modeling is a prerequisite for ISAC system design and has attracted widespread attention from both academia and industry. 3GPP Release 19 initiated the ISAC channel study item in December 2023 and finalized its modeling specification in May 2025 after extensive technical discussions. However, a comprehensive survey that provides a systematic overview,from empirical channel features to modeling methodologies and standardized simulators,remains unavailable. In this paper, the key requirements and challenges in ISAC channel research are first analyzed, followed by a structured overview of the standardization workflow throughout the 3GPP Release 19 process. Then, critical aspects of ISAC channels, including physical objects, target channels, and background channels, are examined in depth, together with additional features such as spatial consistency, environment objects, Doppler characteristics, and shared clusters, supported by measurement-based analysis. To establish a unified ISAC channel modeling framework, an Extended Geometry-based Stochastic Model (E-GBSM) is proposed, incorporating all the aforementioned ISAC channel characteristics. Finally, a standardized simulator is developed based on E-GBSM, and a two-phase calibration procedure aligned with 3GPP Release 19 is conducted to validate both the model and the simulator, demonstrating close agreement with industrial reference results. Overall, this paper provides a systematic survey of 3GPP Release 19 ISAC channel standardization and offers insights into best practices for new feature characterization, unified modeling methodology, and standardized simulator implementation, which can effectively supporting ISAC technology evaluation and future 6G standardization.