No. You won't get the exact same colors with white balance only.
1 - Built in settings
The built-in white balance settings are a very, very generic correction. If you are in a hurry, well, they are better than nothing.
The first basic color correction these settings make is color temperature. But not always the problem is the color temperature, but also, for example, the tint.
2 - Custom White balance
You can make a custom white balance with a grey card to have better specific results.
A white balance is a simple correction between the 3 RGB chanels. Let us imagine that corrects an image using simple curves:
(These graphs are just to get the idea)
Here is an explanation of setting a custom white balance: Color issue: studio images have a pink hue
3 - Profile
But to achieve more controlled color in different light situations you need a profile. A profile makes more elaborated corrections in different zones. Imagine that the camera accurately captures bright blue tones, but darker ones need corrections.
So you need a color chart, not just a grey card. One standard is Macbeth. These charts need to be printed with high-quality pigments, so each patch has a standardized value, that the software can use to adjust curves on the color profile.
The software analyzes the target and makes a profile because it knows what the color should be.
A good color chart is the X-Rite Color Checker Passport. It includes software to make a profile for the combination of your camera + the specific lighting situation in which you take the photo. The color changes can be different from flash to flash, from different brands, different models, or even if one flash is old and the other is new.
4 - Extreme light
But even in some extreme light situations, you can not achieve the same colors, for example, I would not use sodium street lamps for a product shoot. Some types of lighting lack some wavelengths, so some physical colors can not be reproduced.
There are some standards to know this quality of the illuminants, for example, the CRI value. Color Rendering Index.
5 - Dynamic Range
You also need to take into account how much your camera can really perceive inside one single shoot.
That is the dynamic range. And depending on the light situation you have and the settings on your camera you can crop a part of the colors.
This depends mainly on the correct exposure. You can have a good white balance, but if the exposure is not right...
6 - Your camera itself
The sensor and the way your camera processes an image vary, from brand to brand, model to model.
If you add some personalized settings like "portrait", "landscape", "neutral" you can have slightly different colors. Some settings can affect just the jpg or flavors of a RAW file.