Can a deck with a mono-colored commander have cards with other colors than its commander in it?
As the rules you quoted say, the color identity of every card in a commander deck must be a subset of the color identity of the commander, where a card's color identity is the set of the colors of all mana symbols and color indicators in that card's rules text.

There are currently only three mono-white legendary creatures whose color identity contains both red and white:


Archangel Avacyn - color ID WR
General Tazri - color ID WUBRG
Sisay, Weatherlight Captain - color ID WUBRG


The only other candidate that's close to your description and could have been valid is Sylvia Brightspear.  She partners with Khorvath Brightflame, which is red.  In the case of partner, a deck actually has two commanders, and they both contribute to color identity.  It's possible Khorvath never got cast, or you simply didn't notice it was from the command zone when it was cast.

If none of the cases above are true, then your opponent's deck was not valid.