Final answer:
The relationship described is a customer service relationship, specific to hardware support and IT customer support, which falls under the category of Customer Relationship Management (CRM).
Step-by-step explanation:
The relationship described in the scenario where Pike General Hospital is receiving support for their router device through various communication channels such as a support phone line, email, and a Customer Service Portal is known as a customer service relationship. This is part of the wider field of Customer Relationship Management (CRM), which focuses on how businesses manage and analyze customer interactions and data throughout the customer lifecycle, with the goal of improving business relationships with customers, assisting in customer retention, and driving sales growth. In this case, the context is specific to hardware support within the domain of IT customer support.