Adding a second iCloud email account to Mail.app in macOS Sierra
Go to the mail app and open mail preferences. Then, under accounts, hit the + button. Select "iCloud" and input the log-in information. Select ONLY the mail icon. Then hit "Add Account".

By going through the mail app preferences instead of System Preferences you can avoid dealing with all of the iCloud options. Here is a picture of what it looked like when I added a second iCloud mail account. On the left side in the list of accounts you can see two iCloud icons (boxed in red):