Utility is a deliberately obfuscating concept, so it's not surprising that even economists find interpersonal utility comparisons a bit confusing (see the comments on the last link).
The maximization/optimization paradigm (henceforth just maximization) is very useful and seems like it must be true somehow, or at least is the best way to understand economic phenomena. But then the question is, what are people maximizing? The truth is, we don't know. It's awkward and embarrassing to say "people are I-don't-know-what maximizers," so we say "people are utility maximizers." It's a concise and scientific way of expressing our ignorance.
Utility is not happiness. If we thought people were happiness maximizers, we would say they are happiness maximizers. Utility is not pleasure, or satisfaction, or eudaimonia, or anything else. Utility is utility, a code word for ignorance.
That is why interpersonal utility comparisons are impossible. It isn't because utility is ordinal rather than cardinal. That just assumes the conclusion, that utility is something that can't be compared across people. Rather, IUCs are impossible because there is no sense in which Alice can have more I-don't-know-what than Bob.
We can say that Alice has more happiness than Bob. We can say that Bob has more money than Alice. We can say that Alice has more satisfaction than Bob, and we can say that Bob has more pleasure than Alice. All these comparisons are perfectly valid. But I-don't-know-what? Calling that comparison impossible gives it too much credit. It just doesn't make sense.
Many economists don't know this. They talk about how IUCs might be possible in the future as we learn more about the brain, as if utility is a real thing. Utility is not a real thing. It is a placeholder for ignorance.
So IUCs are not even coherent enough to be impossible. Yet there is this intuition that a poor person gets more utility from a dollar than a rich person. Let's explore this. First of all, on most comparisons of any particular value, the poor person gains more than the rich person from the marginal dollar. The poor person gains more happiness, pleasure, relief, etc., than the rich person in most circumstances. If you replace the placeholder of utility with something concrete, comparisons are valid. This isn't an interpersonal utility comparison, it is an interpersonal happiness//pleasure/relief comparison.
Note that you can't always do this. For example, who gets wealthier from an additional dollar, the poor person or the rich person? Obviously neither.
This possibility shows that substituting any particular value for the utility placeholder does not necessarily get us to the intuition that a poor person gets more utility from an additional dollar than a rich person. People necessarily maximize utility. Utility is the thing people are maximizing. But people do not have to be, say, happiness maximizers, and they often are not. So even if utility were a more substantial concept, any particular thing poor people gain more of than rich people do from an additional dollar would not necessarily maximize utility if people are not trying to maximize that value. And since poor people cannot gain more of every possible value than rich people from an additional dollar (e.g. wealth), then simply knowing what specific value utility is a placeholder for is insufficient to generate our intuition that giving a poor person an additional dollar generates more utility than giving it to a rich person.
Here is another problem: people want different things. If the poor person is a happiness maximizer, and the rich person is a satisfaction, then the comparison is again not even impossible. There is no sense in which a dollar can give a poor person more happiness than it gives a rich person satisfaction. The comparison is only valid if both individuals are trying to maximize the same thing.
So this is what we need to do an interpersonal utility comparison: we need to know what utility is a placeholder for, and we need to know that utility is a placeholder for the same thing for both individuals. Having both of these conditions together is unlikely, to put it mildly.
Yet it really does seem that giving a poor person a dollar generates more utility than giving a rich person a dollar. There is one sense in which this intuition is unambiguously and easily true. If you choose to give a dollar to a poor person rather than give it to a rich person, then giving a poor person a dollar does generate more utility than giving a rich person a dollar. It generates more utility for you!
Most people would rather see a poor person get another dollar than a richer person, even most rich people. Maybe that is because we care about happiness, and a poor person is happier from an additional dollar than a rich person (the fact that neither might be happiness maximizers is happily irrelevant). For many other values we likely share, that argument is true. Thus it is true that what is valuable and good is enhanced more by giving an additional dollar to a poor person than a rich person, and this is because the poor person is made happier/more satisfied/whatever than the rich person by the additional dollar, even though the interpersonal utility comparison is impossible.