It's absolutely not legit at all. This is pseudoscientific woowoo, and can potentially be dangerous. As far as it being an attempt to change and perfect your body, this is the same as the weird "mewing" trend in that it appears to do something in a way that might convince you that you're progressing in some way, but the actual thing happening has absolutely nothing to do with your goal. Mewing teaches you to hold your tongue high for a time period in the hopes that it will somehow magically make you have a stronger jawline permanently, but it functionally cannot do anything of the sort. Here, you're just learning to do weird movements and breath-holding, this is in no way improving your breathing.
The only evidence-based outcome anyone has found for Buteyko breathing in the last 70 years is that it can have a slight impact on a couple of asthma symptoms leading to a reduction of need for bronchodilator use without actually improving lung function, but does not affect the cause of those symptoms. If you're not an asthmatic who wants to reduce their inhaler use, science has not substantiated any of the many claimed other positive effects.