Hi Piotr,
Based on my test, I can login to OneNote for Mac with Office 365 Essential account without any problem. Also I tested the Microsoft account. Given the situation, we need to collect some information. So, to protect your privacy, I will send a private message to you for requesting the information, please reply to it in the private message.
To respond to the private message:
1. Please go to your details section on the right side of the community site.
2. Click Private messages.
3. Click the subject title and respond to it.
Thanks,
Dunk