The jpg format is 8-bit only - the whole point of jpg is to produce small files. (16-bit files are twice the size of 8-bit files)
If you want more than 8-bit, you have to export as 16-bit Tiff or PSD.
But you would only need to do that if the image is going to be edited after exporting - otherwise 8-bit is fine.
You can't tell the difference between an 8-bit and a 16-bit image on screen - the advantage with 16-bit is that it can withstand heavy editing without deteriorating - it gives you editing headroom.
For more information about file formats and bit depth - see File formats and What is a digital image?