@Afy,
Office must now be leased, rather than owned. You must sign in with your Microsoft account (the account from which you purchased Office 2016) to access the suite of software.
However, you do not need to store your files in OneDrive.
You may
- Set OneDrive "Settings" to not "Use Office 2016 to sync Office files that I open"
- Store your office files (*.doc, *.docx, *.xls, *.xlsx, *.xlsm, *.ppt, *.pptx, etc.) in a folder other than in or below your "Documents" or "OneDrive" folders, so that you may continue to use OneDrive without syncing Office files.
I hope this helps.