Windows 10 will still let you pick a default browser that isn't Edge, and even if you use Edge or Internet Explorer, it will continue to let you pick a default search engine other than Bing. But in a change which goes into effect today, Web searches that use the Cortana-driven search box in the Windows 10 taskbar will no longer give you any choice: they will always open in Bing, and they will always use Edge to do so.
Microsoft explains that it is making this change because of the smarter capabilities that it has built in to its Cortana digital personal assistant, and the integration this requires of the browser and search engine. For example, the company says that a search for "Pizza Hut" in Cortana will, when opened up in Edge, show locations and directions and rich information. Plans are afoot to make this feature more capable and extensive; eventually you will be able to ask Cortana to "get tickets to Rihanna show," and Windows will find appropriate tickets and streamline a customer's purchase.
This integration requires a common understanding of entities and semantic information about what is being searched for, and how to present this data. Microsoft can offer that in its own platform, but has no consistent, reliable way to do this with other browsers or search engines.