Microsoft SQL Server Connection Coloring
Boiling down the question to the basics:

Q: "is there some way to set a policy or something on the Microsoft SQL Server instance itself that will cause any SSMS clients to be colorized?"

No. SSMS query color-coding is part of a client-side connection and that data is stored in the registry. You could theoretically export that registry setting and apply it to anyone in your company with SSMS, but it's not a trivial task for us database people. 

I'd take that request to your sysadmins with Active Directory Group Policy administration duties and explain what you need. It's a fairly common type of request for companies that manage software deployments centrally. Unfortunately, it's not point-and-click easy, and you'll need to update the registry keys every time you add or rename a production server. These types of mass client deployments usually require change requests because modifying the Windows registry isn't something you want to be doing often.