The relentless proliferation of Big Data and artificial intelligence has compelled computing platform architectures to evolve into heterogeneous multicores for greater energy efficiency. A customized network-on-chip (NoC) supporting interconnection diversity is pivotal for the asymmetric data-access traffic requirements of modern heterogeneous multicore system-on-chip (SoC). A significant portion of on-chip data access comprises single-source multi-destination (SSMD) traffic, which supports barrier synchronization, multi-threading, cache coherency protocols, and deep neural network (DNN) acceleration. By amortizing SSMD traffic, multicast routing is essential for effectively utilizing communication bandwidth. One of the primary concerns in supporting multicast routing in NoCs is to circumvent the additional deadlock conditions caused by branch operations among the active routers. However, it is challenging to implement the throughput-optimized multicast routing in irregular topology-based NoCs because the deadlock conditions become highly complicated, and the Hamiltonian path required to apply the labeling rule may not exist. Two important observations were identified regarding multicast routing in customized NoCs: 1) Even if the NoC lacks a Hamiltonian path, deadlock-freedom can be guaranteed by restricting branch operations to a specific destination. 2) A variable path diversity in a custom topology can be leveraged in routing path allocation and branch. Based on these properties, this study proposes a deadlock-free and throughput-enhanced multicast routing for customized NoC (MRCN). MRCN ensures deadlock freedom by utilizing extended routing and router labeling rules. Furthermore, destination router partitioning and traffic-aware adaptive branching are incorporated to reduce packet routing hops and disperse channel traffic. The effectiveness of MRCN was verified using Noxim, a well-known cycle-accurate NoC simulator, under various topologies and traffic patterns. The simulation revealed that MRCN improved the average latency by 13.98 % and the throughput by 12.16 % under the saturated traffic conditions over the previous multicast routings in customized NoCs.