Can the (point-wise) multiplication of two discrete band-limited functions be inherently aliased?
(1) is it possible that the point-wise multiplication of two discrete, band-limited functions is aliased itself?


Yes.  Or at least it's possible that the result won't fit in a tidy manner into your sampling rate.  Because multiplying two signals together creates energy at different frequencies than the frequencies of the original (this is easiest to do by a single-frequency example: $2 \cos \omega_1 t \cdot \cos \omega_2 t = \cos (\omega_1 - \omega_2)t + \cos(\omega_1 + \omega_2)t$.  If that $\omega_1 + \omega_2$ is to great to fit into your sampling rate, then it'll alias.


  (2) would it be possible to circumvent the aliasing by outputing the 'full' convolution, thereby getting $2−1$ samples back?


Yes and no.  Yes, you could, but to be consistent the result would have to be at a sample rate $\frac{2M - 1}{M}$ times the original.

An equivalent operation would be to upsample both signals by a factor of two and then multiply -- if the upsampling were done in a way that results in correctly band limited signals, once you multiplied them they would fit into the new, bigger bandwidth made available to them by upsampling.