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 $m_{\langle \lambda \rangle}$ appering in the $\langle n \rangle$-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 $(G,*)$-algebras whose multiplicities in the sequence of $\langle n\rangle$-cocharacters of $A$ are bounded by $1$ by showing a list of $(G,\ast)$-polynomial identities satisfied by such varieties.