Edge already has Cortana-specific browser chrome. (Some restaurant pages, for example, will light up Cortana's halo in the address bar and clicking that will open a Cortana sidebar full of action cards like "Create a Reservation".) It's not surprising that the plans are to continue adding Cortana functionality in Edge, and to work to make that hand-off between "Taskbar Cortana" and "Edge Cortana" as seamless as possible.
Theoretically they could probably leverage extensions to other browsers, but making sure users had those extensions installed and handing off to those extensions would presumably be A) a bunch of additional work for the teams involved, and B) threaten to show a bunch of seams between the application boundaries, hurting the suspension of disbelief that Cortana is an "individual" working in our favor, potentially weakening Cortana in the public's eyes.
Theoretically they could probably leverage extensions to other browsers, but making sure users had those extensions installed and handing off to those extensions would presumably be A) a bunch of additional work for the teams involved, and B) threaten to show a bunch of seams between the application boundaries, hurting the suspension of disbelief that Cortana is an "individual" working in our favor, potentially weakening Cortana in the public's eyes.