Yet, somehow, the two are wholly different mechanically.
Monks aren't even "exemplification of their alignment". It's just fluff. It's like saying Bards are the same as paladins because they are the exemplification of the nonlawful alignment.
I've always thought it's funny that barbarians and bards can't be lawful, but rogues can. A lawful character can sneak attack, but never become enraged or (learn to) sing. The rage doesn't even cause loss of control or tactics, just inability to focus on advanced skills or spellcasting. there's a perfectly reasonable exalted barbarian prestige class.
It's also cute that bards can rather uniquely become lawful without losing any of their abilities (barbarians lose the ability to rage). They just can't gain more bard levels unless they're currently a free or neutral spirit. (Oh weird, monks are the same way actually)
Anyway there certainly are similarities between monk and paladin. Both are primarily melee characters who gain disease immunity, limited healing ability, and some extra striking abilities. They are even likely to be trained by similar organizations, monasteries vs holy orders. Their abilities both benefit from strength and wisdom (though paladins focus more on charisma probably).
But there's a world of difference, too. Monk abilities don't help others like paladin ones do. They basically can't wear armor, don't get a mount, don't really get spellcasting, almost have to fight unarmed... Very different.
Monks don't even have a code of behavior, which I guess was probably the jerk's misconception. A lawful character is certainly allowed to break laws, they just need to show a clear preference for order in general. The morality system doesn't require absolute adherence... Except for paladins, who aren't even allowed to associate with people who anyone who "consistently offends their moral code"... Hence their reputation for triggering party fights, sadly, since the rules require them to hold *everybody* to their morality system (though with leniency).