I know what you mean as I upgraded from Windows 7 to Windows 10 when the free upgrade was offered. I had a user account in Windows 7 that transferred over to Windows 10. (but I did have a password) After using Windows 10 for a while I switched to using a Microsoft account but when I have clean installed Windows 10 I always use a local windows account first.
I think your local account is somehow corrupt.
The built-in Administrator account in Windows 10 is there as an emergency, but you need to make it visible. I suspect you cannot do this in the current situation as it requires a command using either a Command Prompt (Admin) or PowerShell (Admin).
Do you have a Windows 10 bootable USB (for windows installation) or a bootable DVD for the same purpose. If not you can create one yourself but you may need another PC/Laptop to create it.
The current version of Windows 10 available from Microsoft via the Media Creation Tool will create a bootable USB or a downloadable ISO (which you can use to create a boot DVD) http://www.microsoft.com/en-us/software-download/windows10
Instructions are on the website about using the tool to create the bootable media. Select Download tool now.
If you can create the media and boot from it then select Language > Repair and you should be able to access the Troubleshoot menu where you can select Command Prompt. At the prompt enter net user administrator /active:yes press return. Restart the PC to see if you have the option to use the built-in account, it should appear on the sign-in screen, there is no password.
I would suggest going to Settings > Accounts and changing your current local account to a user type account then restart using Start > Power > Restart.
Sign-in again and and using the Administrator account go back the Settings > Accounts and change your local account back to an administrator type account.
Restart again and and this time sign-in to your local account and try something that brings up the UAC again, perhaps just Command Prompt (Admin) or PowerShell (Admin).
Failing that you may have to create a new local account and start to use that instead of your previous account from Windows 7 as an administrator.
https://support.microsoft.com/en-us/windows/create-a-local-user-or-administrator-account-in-windows-10-20de74e0-ac7f-3502-a866-32915af2a34d
When you have finished using the built-in Administrator account then turn it off again by going to a command prompt, as before and enter net user administrator /active:no press return, you should not leave this account enabled.