Is there a notation to designate special type of isomorphic relation? Let $R,S$ be some structures. Is it okay to write isomorphic notation like $R\cong_{ring} S$ or $R\cong_{group} S$, if $R,S$ are groups or $R,S$ are rings? Is there a standard notation to designate a type of morphism?

Yes, the standard notation is $\cong$. There is no ambiguity if one is careful: When $R,S$ are rings, then $R \cong S$ means that there is an isomorphism of rings $R \to S$. Don't write $R \cong S$ when you only mean that the underlying additive groups are isomorphic. This is really just $U(R) \cong U(S)$, where $U : \mathsf{Ring} \to \mathsf{Ab}$ is the forgetful functor. (Don't forget forgetful functors.)