In this case, you better learn by heart that 'どちらも' means here 'both of them',
because 'どちら' has different meanings.
'桃' and '緑' belong only to the noun, and for example '赤', '黄' or '茶' are also the nouns,
which has however respectively a i-adjective form '赤い', '黄色い', or '茶色い' at the same time.
So '緑の傘' is the combination of the noun '緑', the particle(jyoshi)'の' and the noun '傘'.
You cannot say '桃色い' or '緑色い', both of which are incorrect.