Can the Edge web-browser by Microsoft run on multiple platforms?

Project Spartan will only support Windows 10.

Internet Explorer 11 will remain unchanged, which is included on non-mobile versions of Windows 10, to provided the required legacy support in an Enterprise environment. Mobile versions of Windows 10 will only have Project Spartan on them. Devices that run non-mobile versions of Windows 10 will have both Internet Explorer 11 and Project Spartan on them.

It should be noted that the linked article in another answer's original revision describes Microsoft's original plan for Project Spartan and Internet Explorer 11. Project Spartan can be thought of as the next version of Internet Explorer without any legacy support. It is very unlikely that Windows 7 or Windows 8.x will receive Project Spartan in any capacity.

The outdated plan has changed. The current plan is described in another article with the following image.

Project Spartan was built for the next generation of the Web, taking the unique opportunity provided by Windows 10 to build a browser with a modern architecture and service model for Windows as a Service. This clean separation of legacy and new will enable us to deliver on that promise. Our testing with Project Spartan has shown that it is on track to be highly compatible with the modern Web, which means the legacy engine isn’t needed for compatibility.

For Internet Explorer 11 on Windows 10 to be an effective solution for legacy scenarios and enterprise customers, it needs to behave consistently with Internet Explorer 11 on Windows 7 and Windows 8.1. Hosting our new engine in Internet Explorer 11 has compatibility implications that impact this promise and would have made the browser behave differently on Windows 10.

Source

Spartan will run only on Windows 10 (desktop, tablet and smartphones). More information.

No. It is the next generation browser engine from Microsoft. Initially released for Windows 10 desktop/tablet but should also be released for Windows 10 Mobile.

The intent is to eventually remove Internet Explorer with its Trident engine. However, this wont happen for a number of years since Trident supports a lot of backwards compatibility that Spartan doesn't and this is required for Enterprise users.