Network-on-Chip (NoC) is widely used to facilitate communication between components in sophisticated System-on-Chip (SoC) designs. Security of the on-chip communication is crucial because exploiting any vulnerability in shared NoC would be a goldmine for an attacker that puts the entire computing infrastructure at risk. We investigate the security strength of existing anonymous routing protocols in NoC architectures, making two pivotal contributions. Firstly, we develop and perform a machine learning (ML)-based flow correlation attack on existing anonymous routing techniques in Network-on-Chip (NoC) systems, revealing that they provide only packet-level anonymity. Secondly, we propose a novel, lightweight anonymous routing protocol featuring outbound traffic tunneling and traffic obfuscation. This protocol is designed to provide robust defense against ML-based flow correlation attacks, ensuring both packet-level and flow-level anonymity. Experimental evaluation using both real and synthetic traffic demonstrates that our proposed attack successfully deanonymizes state-of-the-art anonymous routing in NoC architectures with high accuracy (up to 99%) for diverse traffic patterns. It also reveals that our lightweight anonymous routing protocol can defend against ML-based attacks with minor hardware and performance overhead.