Hey Elzbieta!
My name is Ravi, and I am an independent advisor. I am happy to help you. :)
A user can restore the backup of a computer to another computer only using the same Microsoft used on the first computer. It's because backups are associated with a Microsoft account. It's not possible to automatically restore backups using a different Microsoft account.
>> Would it be possible to transfer the files somehow over from one account to another?
No, it's not possible to transfer files from one account to another.
There are two options.
1). Log into the same Microsoft account used on the first PC.
2). Manually transfer the data to an external drive from the first PC and then transfer it to the new PC.
You can find the information related to back up here: https://support.microsoft.com/en-us/windows/bac...
Feel free to get in touch again if you have any additional questions.
Cheers,
Ravi