Yep, she is. It is a pretty rare type. Primary melancholics in general seem to be a bit rare, though melancholic/cholerics are staples in fiction because they’re so dramatic and make plots move. The other two kinds of melancholic don’t get much representation.
Mai has both of the cold humors, and the coldest one of all as her primary. Because of this, she isn’t particularly active or motivated. Her melancholic side needs ideas and stimulation, but she’s too laid-back and passive to go out and seek them, which is why she’s always bored. The fact that both she and Ty Lee (sanguine/phlegmatic) have some phlegmatic in them is why they were good at following Azula—but also why they turned on her when she went too far.
“Melancholic,” in the humors sense, doesn’t necessarily mean sad, though they are prone to being hard on themselves, that doesn’t mean they’re incapable of being happy or contented. A melancholic is happiest when they have time and stimulation for their ideas, and can spend a lot of time talking, reading, theorizing, etc, with people who understand them, though they’re also a very introverted humor and need some alone time as well. Given that her childhood seemed very silencing, she’s starved for this kind of interaction and would really enjoy it.
Mai needs external stimulation, someone to dredge her up out of her rut. Sanguine antics might entertain her, but she’s going to feel a lot closer and more intimate with another melancholic. She’s highly compatible with Zuko, and both being primary melancholics, they understand and relate to each other very well, while their choleric + phlegmatic sides are attracted to each other, and his keeps things interesting. Being a primary melancholic, though, Zuko is a bit vulnerable to eventually being dragged down into Mai’s rut with her, instead of being the one to pull her out of it. This might not bother them too much, though, as a rut’s not so bad when you share it with someone you love.
Azula is also highly compatible with Mai. They don’t relate to each other quite as well, but they still do very well, and Azula’s power of attraction for Mai is even stronger. Being primary melancholic, Mai is more morally flexible than most people with a phlegmatic humor, but she still has lines she will not cross, and unfortunately Azula crossed them. If taken before The Boiling Rock, though, or afterward in a scenario where Azula is redeemed (very likely with Mai’s help), Azula would make an exciting and attractive partner for Mai.
Not quite as compatible, but a match with a lot of potential is Sokka. He’s sanguine/melancholic, so he’s all quirky and weird in ways that will baffle (but also entertain) Mai, but he’s got that melancholic side which she might find a pleasant surprise. There won’t be the immediate attraction she feels to a choleric, and she won’t ever quite know where she stands with him, but being equals isn’t really a bad thing either.