Under Xen, your instance is not quite a process in the conventional sense. The Xen hypervisor lives underneath all the OSes on the host, both those allocated to customers (domUs in Xen-speak) and the one allocated to manage the customers (dom0). Xen starts up before the dom0 kernel, and then loads the dom0 OS as a privileged instance.

This is quite different from KVM, where the hypervisor is built as kernel modules in the linux kernel, and the host linux OS acts as the management instance.

Functionally, however, the results of this setup are similar to being a process: the hypervisor may schedule your instance's CPU time, and your kernel is specialized so that when it needs memory, it calls into Xen for the appropriate mapping (similar to VM in kernel).

It's also important to note that Xen existed before and works without hardware virtualization support. That is one of the main reasons for it's approach, there was no commodity support for virtualization within the CPU, so the only safe place to handle it was very low level within the kernel.