Putin is doing this for popularity boost, as he has successfully employed several times before. The reasons he gives publicly are contradictory.

And in any case the treaty doesn't allow for invasion upon the government being perceived as illegitimate. This is an aggressive war, considered since the end of WW2 to be the supreme international crime.

The problem still so far is everything done to hold Putin accountable is viewed by Russians as the West vs Russia. This invasion is viewed not as aggressive war, but liberating Ukraine from Nazis. Exactly what Putin says domestically.

And yes, most of the rest of the world recognizes Taiwan as a sovereign country, except formally they use language consistent with One China to avoid a row with China