How to export as PNG with 16 bit depth?

[Ofnuts]

Sorry but this color encoding doesn't exist in the PNG format.

So either:

• Despite the PNG extension the file is a bitmap (same contents as a BMP file)
  
• This is a PNG file. Note that according to the Wikipedia description, the "bit depth" is the number of bits per channel, and not the total number of bits per pixel (color depth) which is 3 or 4 times greater in color images. So what you are showing would be a high precision PNG with 16 bit per channel (instead of the more common 8 bits) and if it is also 16-bits per pixel then it is grayscale without alpha.
  

You can use utilities such as mediainfo or ImageMagick's identify to have a more accurate description of the actual file format.