Why does iCloud ask for my password twice every time I turn on my Macbook?

In the iCloud pane of System Preferences, uncheck the box marked Mail and check it again.

What else have you done that you didn't mention?

Open the Sharing preference pane and look at the Computer Name. Does it end with a number in parentheses; e.g., "Somebody's MacBook Pro (2)"?

Back up all data.

Launch the Keychain Access application in any of the following ways:

• Enter the first few letters of its name into a Spotlight search. Select it in the results (it should be at the top.)
• In the Finder, select Go ▹ Utilities from the menu bar, or press the key combination shift-command-U. The application is in the folder that opens.
• Open LaunchPad. Click Utilities, then Keychain Access in the icon grid.

Select the login keychain from the list on the left side of the Keychain Access window. If your default keychain has a different name, select that.

If the lock icon in the top left corner of the window shows that the keychain is locked, click to unlock it. You'll be prompted for the keychain password, which is the same as your login password, unless you've changed it.

Right-click or control-click the login entry in the list. From the menu that pops up, select

Change Settings for Keychain "login"

In the sheet that opens, uncheck both boxes, if not already unchecked. From the menu bar, select

Keychain Access ▹ Preferences ▹ First Aid

If the box marked

Keep login keychain unlocked

is not checked, check it. Select

Keychain Access ▹ Keychain First Aid

from the menu bar and repair the keychain. Quit Keychain Access.

Are you using iCloud Keychain?

Have you signed out of iCloud since this started? If not, do that now and sign back in. Otherwise, what login items do you have?

Select the Login Items tab in the Users & Groups pane of System Preferences. What are the login items?

And?