Sony A6500 Camera Calibration

I shoot my photos as raw file by Sony A6500 camera. Then, I used Adobe converter to convert the ARW to DNG file ( for some reason my lightroom do not recognize the ARW file).

It depends on the Version Number of Lightroom that you are using. (Minimum v6.8  CC2015.8 needed)

Cameras supported by Camera Raw

However, when using Lightroom to edit them, the camera calibration only have one option "Adobe Standard" instead of Landscape, Deep, Clear, Light,...

Commonly, "Mirrorless" cameras (as is the A6500), the profile is embedded in the image metadata by the camera. Lightroom will only show the Adobe Standard.

The Sony A6500 was first supported by LR 6.8 or LR CC 2015.8 and your LR is clearly older than that.  If you have a LR 6.x series and just need to update then download and install the patch installer for 6.14 from here:

https://helpx.adobe.com/download-install/kb/lightroom-downloads.html

If you have an older version of LR than 6, like LR 5.x, then why?. You have the power to fix it by paying Adobe for their work in supporting your camera.

The current desktop version of LR Class CC is 7.4 and available along with Photoshop as a $120/year subscription which is billed at $10/month, I think, still, unless it's gone up a $1/month, I'm not sure.