Intuitively, I would say that "associativity of the monoidal product" should mean:
for all objects $A,B,C$, there is a natural isomorphism so that $(A\ast B)\ast C \cong A\ast (B\ast C)$, and
for all morphisms $f,g,h$, it holds that $(f\ast g)\ast h=f\ast (g\ast h)$.
Obviously, nr 2 doesn't make sense, given that the objects of $(f\ast g)\ast h$ and $f\ast (g\ast h)$ merely have to be isomorphic but not identical.
Is it true that the coherence conditions imply (or even are intended to imply) something like a weaker version of nr 2? If not, then how do the coherence conditions capture the intuitive notion that the monoidal product is "associative"?
The closest thing to your second condition that we could ask in a monoidal category is that the the two terms of your equation become equal if you compose them with isomorphisms $$\alpha_{A,B,C}:(A\ast B)\ast C \to A\ast (B\ast C)$$ and $$\alpha_{A',B',C'}:(A'\ast B')\ast C' \to A'\ast (B'\ast C')$$ to compensate the fact that their domains and codomains don't match : this would give you the condition $$(f\ast( g\ast h))\circ \alpha_{A,B,C}=\alpha_{A',B',C'}\circ ((f\ast g)\ast h).$$ Asking this to hold for all $f,g,h$ (and the isomorphisms $\alpha_{A,B,C}$ to depend only on the objects) is equivalent to asking that $(\alpha_{A,B,C})_{A,B,C\in Ob(\mathcal{C})}$ is a natural isomorphism, from the functor $(\_\ast \_)\ast \_$ to the functor $\_\ast (\_\ast \_)$, and this is part of the definition of a monoidal category; but that's not part of the coherence conditions.
The coherence conditions are intended to make sure that there is no ambiguity when you're looking for an isomorphism $(A\ast (B\ast (C\ast D)))\to (((A\ast B)\ast C)\ast D)$, for example : a priori you could move the parenthesis in different ways, and you want to make sure that the isomorphism you get at the end is always the same.