Hi Piotr,
This issue may be caused by you use the same email address for both Microsoft account and Office 365 account.
I suggest you setup an alias for Microsoft account, set this as the primary by the steps below:
1. In Outlook.com, click your user name in the upper-right corner.
2. Click Account settings. You might have to sign in again.
3. Click Overview.
4. Click Add or change aliases, and then click Add alias.
5. Enter a new email address, and then click Add alias. If the address you request is already taken, you’ll be prompted to enter a different address.
- or -
Select Add an existing email address as a Microsoft account alias, enter an existing email address, and then click Add alias.
6. Go to http://account.live.com .
7. Click Account aliases.
8. Click Make primary under the alias you want as your primary.
You may get more information from http://windows.microsoft.com/en-us/windows/outlook/add-alias-account.
If the issue persists after doing these actions, could you please change the Office 365 account back to user@contoso.onmicrosoft.com? Then try logging in to OneNote by using this account.
Thanks,
August