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Given two identical particles, Wikipedia says that the wavefunction of a combined system where the first particle is in state $|n_1\rangle$ and the other one is in $|n_2\rangle$ is $|\psi\rangle=|n_1\rangle|n_2\rangle\pm |n_2\rangle|n_1\rangle$.

I assume that this is because you want $|\langle\psi|\psi\rangle|^2$ to be invariant under the exchange of particles. However, why not a more general $|\psi\rangle=|n_1\rangle|n_2\rangle + e^{i\phi}|n_2\rangle|n_1\rangle$?

There is a brief discussion on the fact that there are these exotic "anyons" that (I think) have these wavefunctions but why is it that bosons and fermions are the only common particles observed in nature? In other words, why are $\phi=0,\pi$ special?

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It's because first homotopy group for configuration space of one-particle states in 3D space is permutations group.

In two words: scalar product $\langle \mathbf p_{1}, \mathbf p_{2}, ...| \mathbf k_{1}, \mathbf k_{2} , ...\rangle$ is invariant under simultaneous permutations $\mathbf p$ and $\mathbf k$ for different particles. For identical particles, however, there are in principle different topological ways which lead to a given configuration of states. These ways form classes, the full set of whose creates so-called homotopy group. By knowing homotopy group you can classify all possible statistics for your configuration.

For 3D space, first homotopy group of configuration space is permutations group. This group has two representations, symmetrical and antisymmetrical, who correspond to Bose and Fermi statistics. In summary, the only type of topological ways in 3D space are those who convert $\mathbf k_{1}, \mathbf k_{2}, ...$ into nontrivial permutations of $\mathbf p_{1}, p_{2}, ...$

For 2D space first homotopy group corresponds to the braid group, who has many representations. Thus Fermi and Bose statistics are only partical case of general statistic. So-called anyons are one-particle states in 2D space (for example, on boundary of two materials), and their statistics is not Bose of Fermi one.

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