When I open a RAW file in Photoshop, the white balance information in the photo is lost.

And to be clear, there is no clearly defined color temperature even in the raw photo. The number is just the amount of compensation applied, to achieve a natural-looking color balance, according to various algorithms.

There is nothing that actually measures the color of the light in the scene.

That's why you will also get different temperature/tint-values in different raw processing engines - and that includes the camera firmware. They may all look neutral, but the numbers still be different because they're all different processing engines and different compensation needed.

A rendered file doesn't have a color temperature, while a RAW file does have one assigned as a part of processing. This is why you don't see a temp once you are in Photoshop.

Adobe Camera Raw and the Camera Raw Filter are two similar looking but separate parts of the Photoshop environment.

Adobe Camera Raw works on raw files and allows you to select a colour temperature for the conversion.

The Camera Raw Filter works on open RGB files (which may or may not have been converted from raw files) and the colour temperature adjustment always starts at 0 and allows you to 'warm up' or 'cool down' the image.

If you think you will want to go back to the original conversion and change it then open the image via camera raw as a smart object in Photoshop (to do that Shift +Click on the open button in Camera Raw). Then when you subsequently click on the smart object, Adobe Camera Raw will open (not the filter) and allow you to readjust the colour temperature for the conversion from raw.

Dave