The anonymity of blockchain may be exploited by criminals for illegal fund transfers, thus a conditional privacy-preserving scheme is important for blockchain regulation. Currently, sharding technology under a multi-chain architecture is used to improve blockchain scalability. However, current conditional privacy-preserving schemes cannot work on this architecture. To protect the privacy of the transaction, we present a conditional privacy-preserving transaction scheme (MC-CPPT) for multi-chain blockchain system. In this system, we proposed a zero-knowledge proof based anonymous transaction, in terms of the identities of transaction participants and amounts, which also enables the unlinkability of transactions and indistinguishability between cross-chain and intra-chain transactions in multi-chain blockchain system. In addition, a multi-node regulatory agency is introduced to control the transaction amount and frequency in the system without a single point of failure. Moreover, an ECC-based encryption scheme is proposed to achieve the traceability of suspicious transactions. A security model is defined and the security of MC-CPPT is demonstrated to meet the expected security goals. Evaluating the prototype revealed acceptable performance and additional security features.