Queen did which is good! I'm curious whether King Charles will though as he seems to own a fair number of private enterprises, as well as the other royals (I think Prince Harry and Megan Markle announced raising a Series B for their company earlier today for example)
Yes, King Charles III will pay the same taxes as Queen Elizabeth II did - he agreed to it during the Accession Council meeting. Only the Monarch and income from the Duchy of Cornwall (paid to the Prince of Wales) are exempt from income, capital gains, and inheritance tax.
Every other member of the Royal Family are liable to tax, the same as us. They can offset the cost of their official duties against their taxes though.
