The strength of a claim will depend greatly on need. In case of limited resources, the one with the greatest need has the strongest claim. In food terms, for example, if there is one bowl of rice left and everyone's satiated while one person is about to starve, their claim on the rice bowl is the strongest, so they are awarded the rice bowl. Someone with an emotional desire can cry about that bowl of rice all they want, even if it was their rice, the starving person is completely justified in enforcing their claim.

Even without need, humans still want things. It's built into us in the form of emotions. This is just human nature and a good moral system is built around our emotions.
However, letting emotional desire alone dictate who gets what would mean that all you have to get everything you want, even over the need or wants of others, would be just crying loud enough until you get it - and we are not small, spoiled children (or at least we shouldn't be). While need is an obvious reason for a claim - maintaining survival of a human being for example - emotional desire is more than just need. Emotional desire involves wanting things even if we don't necessarily need them. These kinds of claims are valid and reasonable (because even just wanting to survive is essentially just a desire in a human brain) - but without an actual need to compare, this type of claim is a different type of beast. A beast to be tamed using fairness. We “offset” a person's emotional desire for something by fairness.

For example, every kid at the table wants some of the tasty bacon. There is enough food for everyone, so everyone is going to survive, but the bacon specifically is limited (and coveted), so we need to set priorities beyond just the need to survive. So, assume a situation which one kid didn't have any bacon yet while another wants a second portion of bacon. There's only one portion left, so whoever gets that last bit of bacon will be the last to get any bacon, the other will go home without. Maximum fairness in this situation is achieved by letting the kid who didn't have any bacon at all, yet, have the last kid of bacon. That way, everyone had the same amount of bacon and nobody was “preferred” over anyone else.

This system alone may depict a fairly communist view of the world. And the fact that this is my personal belief system and that I agree with it may be a sign that I should also be in favor of Communism, which I may or may not be.¹⁾ Either way, while this system may be nice and fine on paper, this obviously won't play out quite as well in reality, mainly because of two things: Ownership and Physical Restrictions.

I respect, to a degree, the concept of ownership. I believe that ownership is not a physical, tangible thing - it's not a fact of reality - but a result of emotional processes in our brain. When creating or otherwise “fairly” acquiring a thing, we tend to grow emotionally attached to it. Quite a lot, actually. I believe that this essentially amounts to a valid²⁾ Emotional Desire claim on that object. In a world where resources are limited, the loss of a thing potentially very damaging and acquiring a replacement very hard, this can create very strong emotional claims on things and is the reason why possession was legally (and morally) codified in the first place. Limitation, Damage and Reacquisition are the fundamental principles leading up to the concept of “ownership”: Breathing “someone else's air” doesn't exist as a tangible problem until air is scarce, the repercussions of running out of air severe (asphyxia, lethal) and reacquisition of air very difficult (think of a spaceship, for example).