By neutral color, I meant target color of white balance (or color of white after correct white balance).
'White' doesn't have a color temperature. The light needed to make something look or reproduce as white has a color temperature. Light of any color temperature can be made to look white in a photo. It can also be made to look orange, blue, red, or any other color we wish to make it look by adjusting the amplification of the red, green, and blue channels in the image we have taken under that light. We call the total channel amplification for the three color channels in photographs the white balance. Color temperature is one axis within white balance that runs from blue to yellow and corresponds to the color emitted by a black body radiator at a given temperature measured in degrees Kelvin.
The adjustments we make between the raw information collected by the camera and the photo we want to end up with that makes something look white is not a color temperature per se, it's a compensating filter that adjusts the relative strengths of the red, green, and blue components in the picture so that the red, green, and blue values are equal for the objects we wish to appear white. We assign a color temperature number to a certain set of multipliers because it is the appropriate one needed to compensate for a photo that was taken under light that was centered on that color temperature. Please note that at any particular color temperature setting, we may also need to alter the green←→magenta axis setting that runs roughly perpendicular to the blue←→yellow axis on a color wheel in order to make a particular object look white. This is because not all light sources emit light that falls exactly along the color temperature continuum defined by the temperature, in degrees Kelvin, of a black body radiator. For example, the LED lighting currently used for stage illumination in a lot of small night clubs can have a much more magenta tint to it than a black body radiator will emit at any temperature. Typical old style fluorescent lights, on the other hand, emit a much greener tint than a black body will radiate.
When we alter the color temperature setting of a photo we have taken, we don't change the color of the light that was present when the photo was taken. Rather, we change how much each of the RGB channels is amplified compared to the other two RGB channels.
A color temperature setting is a set of multipliers for the red, green, and blue channels that is appropriate to apply to a photo taken under light of a specific color temperature. This affects what color various objects in the photo will appear to be, but it doesn't change "their color temperature" because those objects do not have a color temperature - the light that was illuminating them has a color temperature.
If we photograph a white object under light that is 2700K, we need to apply a 2700K color temperature setting for that object to look white in our photograph. If we photograph the same object under light that is centered on 8000K then we must apply a color temperature setting of 8000K for the object to look white in our photograph. If we apply RGB multipliers (i.e. a color temperature setting) appropriate for 5000K light to the first image taken under 2700K lighting the white object will look blue, if we apply RGB multipliers appropriate for 5000K to the second image that was taken under 8000K lighting the white object will look yellow/orange.
What's the color temperature of neutral light?
There is no such thing as neutral light. I think what you are trying to ask, though, is what is the color temperature setting to make the light illuminating a scene appear to be neutral in the resulting photograph.
In that case the appropriate color temperature setting to make a specific color of light appear to be neutral in a photograph will always be the setting that corresponds to the color temperature of the light under which the photograph was exposed. More correctly, the white balance setting should match the total white balance of the light used to illuminate the scene.
If what you are trying to ask is instead more about the color of light needed to illuminate a photograph to make the white objects in the photograph appear to be neutral, then the color of the light illuminating the photo should be the same as the color to which the system used to produce the photo was calibrated. If you edited a photo on a monitor that was set to D50 (broad spectrum light centered on 5000K) then you would need to use lighting that conforms to the D50 standard to make a print look as close to the same as is possible when viewed as the photo looked on your monitor. If you edited the photo on a system set to D65 (broad spectrum light centered on 6500K) then if you view the image on another monitor it would also need to be set at D65 for the image to look the same.
I think some of your confusion may be understanding what color temperature means when talking about the light illuminating a scene when we take a photo as compared to what color temperature means when talking about calibrating a display system. The two are certainly related, but they are completely different aspects of the same thing.
If we take a picture under 3500K lighting and apply a color temperature setting of 3500K when we convert the raw data to an image then the photo is said to be properly balanced.
If we then view it on a system that complies with the D65 standard when we are in a viewing environment that has D50 ambient lighting, the colors will not appear to be neutral. White objects in the image will appear to be blue/cool to our eyes because the warmer D50 ambient light has acclimated our eyes and brains to see D50 light as "white." Conversely, if we view an image produced on a system set to D65 in a viewing environment that complies to the D50 standard, the white objects in the photo would appear to be yellow/warm.
In the end, to get 'neutral color' we must apply the same white balance setting to an image as the white balance of the light that illuminated the scene when we photographed it. Then we must view the image under the same color of ambient lighting as the color to which the system that reproduced the image was calibrated.
My question was what is the target illumination color and its temperature for white balance in a camera?
There really isn't one and there doesn't need to be one.
Here's why: If we have an image that has an RGB value of (235,235,235) at a specific spot in the image that we expect to be white then that spot in the image can be said to be white (very light gray actually, pure white would be (255,255,255) but that opens up a whole 'nother can of worms).
When that (235,235,235) RGB value is displayed on a monitor, the system displaying it transforms the (235,235,235) value to whatever value the color profile for the system says it needs to send to the monitor for that spot to look white on the screen. Likewise, if we print that image using a high quality printer that has been properly calibrated with the computer we are using to print it, the printer driver will translate that (235,235,235) RGB value to specific amounts of each color of ink to result in the correct color on a particular paper. If we change the type of paper and the printer is properly informed of what type of paper we are using, the amount of each color of ink it uses will change for the same (235,235,235) RGB value in our photo.
As long as the color profiles used to display the image, whether it be on a monitor or in a print, are correctly calibrated for the viewing conditions (ambient light, type of paper, properties of the printer ink, etc.) when we actually look at the monitor or print, then the (235,235,235) RGB value will continue to appear neutral when we look at it. If there is a mismatch at any point in the process, then we will no longer see the point with a (235,235,235) RGB value as rendered to be neutral in color.
The white balance setting (which includes the color temperature setting) must match the white balance of the light that illuminated the scene when we took the picture. If we do that properly neutrally colored objects in the scene will have RGB values that are neutral. The red, green, and blue components used to represent the neutrally colored object will be equal to one another in the raster image format (e.g. JPEG) used to store the image.
The system profile setting (e.g. D50 - broad spectrum light centered on 5000K or D65 - broad spectrum light centered on 6500K) must match the viewing conditions in order for that neutral color to be maintained when the raster image is rendered by a monitor or printer.