User mode Linux does actually run something like a real kernel in user space. OP is just providing a POSIX API with no kernel implementation whatsoever. It just forwards those calls to the Window's API. The library OS concept linked to above is different again – that is an actual kernel running in kernel mode but as part of a unikernel (kernel code and application code linked together into a single process).

Still pretty cool but overshadowed by the claim that they've written a whole kernel.

I see. Thanks!