The reason that a $\pi_0$ can’t decay into one photon is that a massive particle decaying into a single massless particle violates the conservation of momentum. Think about it in the rest frame of the pion.

It can decay into two photons because this is compatible with momentum conservation. In the pion’s rest frame, the two photons come out in opposite directions, with zero total momentum. There don’t have to be two kinds of quarks inside for it to do this. For example, charmonium ($c\bar{c}$) decays into two photons, as does positronium on the leptonic side.

It can also decay into three, or even more, photons, but the probability of each extra photon is reduced by a factor of $\alpha\approx1/137$, the fine structure constant. (There can be other numerical factors also.)