Are there ethanol and methanol analogues with more than one hydroxyl group?
Generally speaking, they're called diols when bearing two hydroxyl groups. If they're on the same C, this is called a geminal diol, if they are on neighboring Cs, they're called vicinal diols. The C on geminal diols has a higher oxidation state than that of an alcohol, i.e. which is why they could be seen as relatives of an aldehydes (or ketones).
In a diluted aqueous solution, formaldehyde $\ce{H2C=O}$ actually predominantly exists in its geminal diol form $\ce{H2C(OH)2}$. This is no longer true for 1,1-ethanediol (the hydrated form of acetaldehyde).

The simplest vicinal diol 1,2-ethanediol is better known under its trivial name ethylene glycol (a common ingredient of anti-freeze formulations).

Triols with the hydroxyl groups on the same C (with the oxidation state of carboxylic acids) are generally unstable.