Some cameras have a section in the 'maker notes' section of the EXIF info that includes such information, but far from all cameras do. Even with the ones that do, not very many applications that show EXIF data will display that information in the EXIF data. Some raw convertors will open raw files using that value in the EXIF info as the default value if the selected options for opening raw files specifies such (as opposed to an option selected to open all files with a preset value such as 5200K).

Among the tools I routinely use, it isn't displayed. But if I upload an image taken with my Canon EOS 5D Mark III to Flickr and allow the EXIF info to be displayed, anyone can click on the "Show EXIF" link and then scroll way down to see it.

Here's a piece of a screen shot from a portion near the end of the EXIF info for this image at flickr:

Notice that the "Color Temperature Auto" and "Color Temperature Measured" values are identical. That indicates that when the camera is set to "Auto WB" the measured color temperature/white balance is used.

But keep in mind, color temperature is only a single axis across the entire color wheel we call White Balance. Artificial light sources are often well off the color temperature axis that is basically defined by the color of black body radiators at different temperatures. For instance, in addition to having a color temperature of about 3700K, traditional fluorescent bulbs also emit a green tint along the green←→magenta axis and need correction in the magenta direction. On the other hand, many of the popular LED stage lights found in small clubs are also at about 3700K but also have a decidedly magenta tint that requires compensation in the green direction along the green←→magenta axis. Both types of light are the same basic color temperature but look very different without compensation on the green←→magenta axis that is approximately perpendicular to the blue←→amber color temperature axis.