white balance is at the RAW level a set of 3 multipliers (often just 2 in practise). The linear data from the RAW file needs is R and B channels multiplied to be ‘in line’.
Doing that through ‘temperature / tint’ is actually not that smart if you think about it, but people are used to it. But yes, RAW converters are calculating R/G/B multipliers (of which G often is just 1.0) to temperature / tint, and those calculations are not the same (for good reasons).
White balancing is one of the first steps in the entire RAW pipeline. If you pick a setting that boosts the R and B channels by a lot, it kinda makes sense for the exposure to change (since exposure is also just a multiplication of linear RAW data). Some software tries to normalize the exposure between WB changes, some others don’t.
Since the RAW data is then pretty quickly fed to an camera input profile, and somewhere down the line there is an exposure setting… and exposure affects how we perceive colors with our eyes (as in, dark blue can look quite different in the shadows vs brightly lit), this all isn’t as easy as you might think .
Since every software handles color and ‘color science’ differently, how WB affects exposure and if that is wanted or not and how it affects the camera profile and the color reproduction… is all different software to software.