Windows Network and Sharing Center
...well, the first word(s) that come to mind when comparing the Windows 10 version of the Network and Sharing Center to past iterations is neutered / spayed (IE, it's been Appled Up and is missing stuff). The Windows Setting, Network & Interface GUI is so much of a joke already, it isn't worth making fun of. Quick technical side note though: If you're using a teamed NIC (Combining two NICs into one, sort of) from Intel, Broadcom, etc, the new Windows Settings GUI doesn't even recognize a teamed NIC. See, the joke continues as they don't even follow their "best practices".
Use some powerful tools like the registry editor or PowerShell and the Network Name and Type can be changed.
Here's the location to go to: HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows NT\CurrentVersion\NetworkList\Profiles
From there it can be a bit of a challenge to find the right Key. Go through all of them and look for a DWORD value named ProfileName. Change that to anything you want. The DWORD value named Category can change it from Public to Private and vice-versa (0=Public, 1=Private, 2=Domain)
To change the name of a "network", go to this Hive Key: HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows NT\CurrentVersion\NetworkList\Profiles and hunt for it in a subkey, DWORD named ProfileName.
And since you won't be able to locate the "Unidentifed Network" in the registry, use powershell (see below)
Open it / Run it as an Administrator otherwise it'll cough up a hairball.
Use this command to list the interfaces configured on a computer: Get-NetConnectionProfile
Use this command, combined with the information garnered with the above command to change it from Public to Private or vice-versa: Set-NetConnectionProfile -InterfaceIndex WhatEverNumber -NetworkCategory [Public/Private] (use the word Public or Private without the [ ]s)
Advanced Sharing Settings
And don't forget to check those pesky Advanced Sharing Settings too. Because of course windows will secure everything for Private networks, and open it up to the world for Public networks. Funny to say, but Windows can do this.