If I shared an Excel file from Office 365 that has some Office tabs used, would someone also with Office 365 be able to see those tabs? Or is it if both parties had Office Tabs installed on their computers would both be able to see them?
Office 365 does not support opening multiple Office documents in a single tabbed window. So, you have to have Office Tab installed to see multiple tabs in one window.
If both parties had Office Tabs installed and you sent those file to them, they will of course see the tabs. However, if you have changed the tab color, font size, or other style on a tab, others will not able to see it.