I understand that this is a minor thread hijack, but I was thinking of how I'd deal with this question given giants in my own world, and I actually felt it would work out well. Giants in my world are played with slightly different flavor text, generally created in what, in my own warped mindset, passes for realistic consequences of the existing creatures.
See, in my world, giants are polite. Really polite. By giant standards. They're big creatures that take up a lot of space, use a lot of resources, and can dish out an absurd amount of damage. Any group of giants together has to be pretty polite, because they all need so much food that they either have to work out agreements about who hunts down deer and who goes after the bears or whatever, or else they run out of food pretty rapidly. Unless D&D giants have magical metabolisms or some other magical wonkiness (like, say, if a cloud/storm giant could take a normal-sized chicken-wing and hold it and immediately turn it into a gigantic chicken-wing by his inherent giant-y magic, which makes him and anything he eats magically big), they need to cooperate in order to maintain the cultures that their descriptions say they have.
So, giants in my world have pretty restrictive etiquette. They are required to offer a visiting giant hospitality (although said giant must not despoil their land or hurt their ability to gather food). They must follow proper forms when making a play for power. In fact, given giants' general bloodthirsty and combat-happy natures, I built some restrictive customs into most giant social functions. For example, when I, as a giant, meet another giant, my appropriate greeting is, when translated into common, "I choose not to kill you at this time." This is usually followed by compliments to the other giant, noting how strong he looks, what fine armor he is wearing, and how difficult the fight would be if I decided to kill him -- although I would undoubtedly prevail.
The other giant politely disagrees, noting that I am massively thewed and wielding well-crafted weapons, but that he would anticipate victory going to him should it come to blows. Once it is established that we both feel confident in our victories but have agreed not to kill each other, we can get down to business and trade or share gossip or whatever.
In such a culture, it is perfectly natural that the cold and frosty weapons we Frost Giants use to help us hunt game would be our primary weapons. If we did decide to fight another Frost Giant, such a contest would either be unarmed or using Frost weapons that really didn't do much. Using a Flaming weapon would be, beyond the obvious discomfort I would likely suffer, an obvious indication that I consider myself too weak to fight another Frost giant with only my own strength and prowess. I might have a flaming weapon hidden somewhere, for use when we hunt white dragons or when the tribe agrees that one of our number has committed some truly egregious crime and has refused to pay the blood-debt to the victim's family -- such a weapon is a tool of war or execution, however, and by no means my normal weapon.
If I were fighting a Fire Giant, however, our customs are different enough that I would likely be fine using a Frost weapon, just as he would be fine using a Flaming weapon. The weapons, extensions of our natures, would be considered perfectly justifiable in honorable combat, and would not cast any doubt over the prowess of the victor.
I didn't design this world with "Frost giants use frost weapons" in mind, but it certainly fits into my world without trouble. Possibly just convenient flavor text, but it works for me.
