Windows 10 will allow you to use a default browser other than Edge, and even if you use Edge or Internet Explorer, will continue to facilitate the use of another engine search other than Bing.
But in a change coming today, web searches using the Cortana drawer on the Windows 10 taskbar will not offer any other option than Bing and Edge. Let's say Cortana from Windows 10 won't use any search engine or browser other than Bing or Edge.
Microsoft's explanation is that it is making this change it is due to the 'smart' capabilities it has created your personal assistant Cortana, and the integration it requires of the browser and the search engine. The example he gives is that if you search for "Pizza Hut" in Cortana, when they open in Edge, it will show the locations, addresses and enriched information. The plans are for this feature to be more versatile and more extensive so that Cortana can be asked to buy tickets for a Foo Fighters concert and Windows will find the appropriate ones for the customer to go directly to the purchase.
This integration requires the understanding of some entities and semantic information for what is being searched and how to present this data. Microsoft may offer it on its own platform, but does not have the way to do it in others search engines or browsers, so it relegates that space in the Windows 10 taskbar to the explicit use of Bing and Edge.
Surely this will not increase the use of Edge and Bing, although to tell the truth, Microsoft has already reported that the use of its search engine has grown since integration with Windows 10. By integrating it more deeply, it may increase its use, but others will forget that space in the taskbar for certain searches.