@Tim G 42 Thanks for reaching out!! To manage the on premise servers, I would suggest you to take advantage of the Azure Arc for Servers which allows you to manage your machines which are hosted outside of Azure same as the Azure machines. For more information about the Azure Arc for Servers , kindly refer this documentation.
After onboarding On premise machines using Azure Arc for Servers , you can use Azure Policy or Azure PowerShell to update the tags for the machines.
Hope this helps!!