Thanks for looking into this.
After doing a test again, I do find the EXR is saving 16 bits per channel but PNG and TIFF are not.
You will need Photoshop or an application that can use the Photoshop filter format for this because a specific plugin is required.
Use Count Colors at Telegraphics - Free plugins for Photoshop & Illustrator...and other software
Count Colors does not work with the full bit depth of the EXR image so I had to convert it to 16 bits per channel in Photoshop
The Telegraphics plugin is the ONLY way I have found to count unique colors in images with 16 bits per channel. GIMP, Irfanview, Faststone may show the wrapping format is higher bit depth but the reported unique colors indicate the data is only 8 bits per channel when the image does contain 16 bits per channel data.