Mobile support almost done. I think build for Android will available in this month. It is already working, Vulkan support is still remains to add.
Screenshot by the link: https://www.neoaxis.com/neoaxis/roadmap
Done with OpenGL ES, but it too old. We will skip OpenGL ES support.
.Net Core runs natively on linux and mac as well. Not sure about the mobile platforms.
.NET Core the framework does, but there are other major considerations for game engines, such as the render pipeline. For instance, supporting Mac means you need to be able to render to Metal. This engine appears to support Windows only right now.