I believe that in Mario lore they are explicitly the same character - either her name is “Peach Toadstool” (i.e. its surname and first name), or the
Princess Toadstool is a title - like how the Prince of Wales, Queen Mother, and Princess Royal are all titles in the contemporary British monarchy.
"In Japan, her name has always been Princess Peach (ピーチ姫 Pīchi-hime), but in the West, she was originally known as "Princess Toadstool", due to Nintendo of America renaming her when localizing Super Mario Bros., feeling that "Peach" was irrelevant to the theme of the Mushroom Kingdom."
