What is the functional group of an alcohol and a phenol?

An hydroxyl group, $O H$.
The common group of ethyl alcohol, ${H}_{3} C - C {H}_{2} O H$, and phenol, ${C}_{6} {H}_{5} O H$, is clearly the $O H$ functionality. This function confers a degree of water solubility on the organic molecule; especially so in the case of ethyl and methyl alcohols, which are infinitely miscible with agua.