Algebras with additional structures and multiplicities bounded by a constant

$G$ be a finite group and $A$ a $G$-graded algebra over a field $F$ of characteristic zero. We characterize the varieties of $G$-graded algebras such that the multiplicities $m_{⟨\lambda ⟩}$ appering in the $⟨n⟩$-cocharacters of $A$ are bounded by a constant, in terms of $G$-identities. If $A$ is endowed with a graded involution $\ast$, i.e. if $A$ is a $(G,\ast )$-algebra, we characterize the varieties of $(G,\ast )$-algebras whose multiplicities in the sequence of $⟨n⟩$-cocharacters of $A$ are bounded by $1$ by showing a list of $(G,\ast )$-polynomial identities satisfied by such varieties.