My only theory is bro science tier idea that fat accumulate some toxins and then it released during weight loss.

Anecdotal evidence says this is a valid theory. I personally caused a shingles infection when I was breaking up intramuscular tissues with a lacrosse ball. As well, most massage therapists will have stories about “toxins coming to the surface” for clients after being worked on (especially in first-time clients) (Source: I’ve talked to about a dozen massage therapists about this).

They may say that, but it's unsupported by science. The red flag is that nobody says anything more specific than "toxin".

Personally I feel that “toxin” is appropriate language in the massage industry: the actual individual effects can be widely varied, clients understand it, and it prepares them for potential symptoms without planting any negative biases that would affect their actual experience through placebo effect.