Let $A,B$ be sets and $G,H$ be order-relation for $A,B$ respectively.
Say, i proved existence of an isomorphism $\psi$ such that [$(x,y)\in G$ iff $(\psi(x),\psi(y))\in H$]. However, "$A\cong B$" gives no information about which order relations these sets take.
Is there a notation includes information about given relations?
In the expression $A\cong B$, it is possible to indicate the underlying structure under which $A$ and $B$ are isomorphic. In your example, you could write $$(A,G)\cong (B,H).$$ If, for example, $A$ and $B$ were isomorphic as additive groups, you could write $$(A,+)\cong (B,+).$$
Often the order is implicit. We fix it in the beginning of the discussion to be a particular order on the set. For example, $\omega$ is a set, not an ordered set per se, but when we write $\omega$ we immediately mean $(\omega,\in)$. So writing $(P,\leq)\cong\omega$ is meaningful, despite not specifying the order.
In the case where $A$ has a very particular order, $\leq_A$ and $B$ has a very particular order $\leq_B$, and you don't discuss other orders of those sets, it might be okay to write $A\cong B$ (might be, because you may have written the text clearly, or you might wrote it in a less-clear way which makes the implicit orders obscure).
If you want to be exact, then as azimut writes in another answer here, a common notation would be: $$(A,\leq_A)\cong(B,\leq_B).$$
