The Internet of Things (IoT) paradigm serves as an enabler technology in several domains. Healthcare is one of the domains in which the IoT plays a vital role in increasing quality of life. On the one hand, the Internet of Healthcare Things (IoHT) creates smart environments and increases the efficiency and intelligence of the provided services. On the other hand, unfortunately, it suffers from security vulnerabilities inside and outside. There are various techniques used to identify, access, and securely manage IoT devices. Additionally, sensors, monitoring, key confidentiality management, integrity, and sensitive data accessibility are required. This study focuses on the IoT perception layer and offers a comprehensive review of the IoHT or the Internet of Medical Things (IoMT). The paper covers the current trends and open challenges in IoHT device authentication mechanisms, such as the physically unclonable function (PUF) and blockchain-based techniques. In addition, IoT simulators and verification tools are included. Finally, a future vision regarding the evolution of IoHT device authentication in terms of the utilization of different technologies, such as artificial intelligence, cloud computing, and 5G, is provided at end of this review.