Most people who think they are fours seem to be sensitive people who don't fit in and live very withdrawn lives. It's also common that people equate their wallowing in their emotions for being 4s.
Mistyped 4s usually come about from them making a very basic incorrect judgement of what the other types stand for, that they don't believe fits them:
1 - Too dutiful.
2 - Too focused on others.
3 - Too shallow.
5 - Too emotionally detached.
6 - Too much about trusting authority.
7 - Too happy/not introspective enough.
8 - Too confident.
9 - Too relaxed/accepting.
The most common mistypes are 6 and 9. Sixes are portrayed terribly in most descriptions, they're described as people who have to rely on someone else to tell them what to think otherwise they can't function and are basically portrayed as people who need predictable boring lives and aren't curious, they're just anxious crazy-people who have to fit into a group otherwise they'll always feel worthless.
In essence they oppose the nature of individuality.
Here's a description from this site about Four/Six mistyping:
Fours and Sixes
Fours and sixes can seem similar because both are emotional, self-doubting and insecure. There are some fundamental differences. Fours tend to doubt themselves, but look outside themselves for ideal love, whereas sixes doubt themselves and while they look outside themselves for authority, they also tend to mistrust the good intentions of their chosen authority. Also, fours’ sense of identity revolves around their being different, sensitive and troubled whereas for sixes, their identities revolve more around being loyal and reliable, as well as the standards of their community. Although social fours can be drawn to counter-culture groups, they tend to feel their difference from that group acutely, often experiencing a conflict between their desire to belong and their desire to maintain their individuality. Sixes have an easier time forming bonds within the group, ingratiating themselves more quickly. Both fours and sixes may feel like outcasts, but for fours this is both a source of anguish and a source of pride, whereas it's less tolerable for sixes. Also, while sixes may identify with the underdog, they also can become persecutory themselves by what they perceive as a threat. Fours are unlikely to persecute outsiders to help their social standing. Both can be idealistic, but sixes are more likely to see things in terms of "us versus them" whereas fours tend to find that such partisan stances compromise their individuality too much. Finally, sixes tend to feel anxious in their own mind, whereas fours tend to retreat into fantasy and imagination.
This seems to inflate the desire to be a four over a six in my opinion. Fours are described as troubled idealists, whereas sixes are described as sycophants who'll screw people over to improve their social standing. The persecution makes them sound even crueler, they're not given much integrity here which doesn't make it hard to see why some people would choose being a four over a six. Not to mention that this gives the mistaken idea that sixes have an easy time forming bonds, which is unlikely to resonate with people who feel uncomfortable and out of place in social situations.
As for nines mistyping as fours, nines are often portrayed as peacemakers who are generally happier and refuse to dwell on negative emotions. So they might think they're a four just because they're depressed. They're not exploring their emotions for the same reasons that a four would, but they perceive it that way, because fours are generally described as having a need to experience emotion and not being able to compartmentalize it, which seems more fitting to a type nine who's become depressed enough that they can't block out negativity.