hi amiri,
to narrow down this issue, i’d like to confirm the following information:
- are there any other versions of office co-exists in the two user’s computers? if yes, which version of office is the default program?
- how do you install the office 2013, from office 365 portal, or other way?
- have you tried in another computer with the same office 365 account to see if this issue persists? this will help us confirm if it’s related to the user accounts themselves.
- could you provide us with the screenshot of the extra activation page? to upload screenshots, you can click
use rich formatting and then click the icon of insert/edit media.
to further troubleshoot if this issue is related to the sharepoint library, could you help us confirm the following information?
- does the sign in dialog box appear when you try to save a document to the sharepoint library?
- does this issue occur when you try to use another browser?
- to access the online team site library from office, you can refer to the following steps:
1). open the office client, and select account;
2). add a service to the office 365 sharepoint;

3). saving the document to the sharepoint, please refer to the screenshot below:

- if this issue still cannot be fixed, please go to file > account > sign out of all the accounts, and then sign into the account using the same as the one of the office 365 team site to see if there’s any improvement.

if there’s any update, please let us know.
thanks,
kerwin