security of multiplicative secret sharing

I was having a read of this question: "Is multiplicative secret sharing secure?" posted in 2013 which asks if the simple multiplicative scheme is secure. Everyone agreed that it was information theoretic secure, however I am not so sure that this is the case. In particular, if the secret is 0 then does that not mean that a participant's share is 0? If this is the case than a participant with a share of 0 instantly knows the secret is 0.

Am i missing something here?

And is this the correct way to post this or would I be better off leaving a comment on the post in question?

There is no 0 in the multiplicative group $\mathbb{Z}_p^*$. By the definition of a group, every element must have an inverse. Thus, the unity in a multiplicative group is 1 and there is no 0.