I was expecting it to work like it did on Win8.1
It does... or at least mine does.
When I closed the browser in Win10 it also closes Outlook.
I tried, but I can not make this happen, and I don't understand how a "browser" opening and closing could impact a separate program like Outlook.
In my case, I've chosen Internet Explorer as my default browser and when I click a link in an Outlook email, that is the browser that opens, and I can close the browser and Outlook is not affected. The open email stays open and the Outlook Program remains open as well.
Much the same way it has always worked in my experience.
Which browser opens for you when you click an email link in one of your open emails?
I've not come across it before but perhaps there is some setting somewhere in the browser as to how to handle external content and the program responsible for providing that external content (in this case Outlook).
I guess, bottom line, it's possible that some recent Windows 10 update or upgrade may have impacted your Office 365 install, or maybe even just Outlook.
Microsoft does provide a Support and Recovery Tool which may help repair whatever might be misbehaving.
https://diagnostics.outlook.com/#/Download?env=Outlook