The DHCP Server is present in the Azure Virtual Network (VNet). VNet hands out private IP addresses to resources. You can change the private IP address of a VM by editing the configuration.
Creating your own DHCP Service on an Azure VM is not supported as you stated.
Everything you need to know about DHCP / Virtual Network is below on the FAQ:
https://video2.skills-academy.com/en-us/azure/virtual-network/virtual-networks-faq
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