The needs that we have in relationships aren't of the same variety as those in Maslow's hierarchy. Where we food, water, and shelter to survive, we don't need a specific partner to survive.
The needs we speak of when dealing with relationships are of a different variety.
I have a need for intellectual stimulation, as do the majority of people. Exactly what form (or forms) that takes differs a great deal from person to person. So, the people we choose to interact with to get the specific form of stimulation we need will necessarily vary.
Now, needs of that sort can be met in romantic or platonic relationships. Where romantic needs can be said to be met lies in the intersection of where some of Maslow's hierarchy meets personal relationship. Maslow highlights the needs for sex and security (I think I'm still in the realm of Maslow's work), which can arise from romantic ties.
Those are needs that are addressed (if not fully met) by a romantic tie. In addition, a need such a specific type of intellectual stimulation can be met by a partner. A need for a specific type of interaction is met by a romantic partner--one not met by a majority of other people.
That, I believe, is what we speak of when we speak of somebody meeting our needs. That person fulfills not only the basic needs addressed by simply having a romantic partner, but also addresses (or fulfills) other needs--needs for specific characteristics of interaction.
A need I have in a relationship, for example, is that I have a certain measure of independence. My wife fulfills that need, neither wanting to cling closer than I'm comfortable with nor wanting more distance than I'm comfortable with. The need for a particular feeling of distance/closeness is one that she fills, so it's a need of mine that she addresses. She addresses a great many such needs or she wouldn't be my wife.
And I don't expect her to address all my needs, nor do I expect that I can address all of her needs, which is why we're poly.