If I understand the definition correctly, it seems to me that all the monads I know of on the category of sets can be viewed as strong monads in a natural way.
For example, if $T$ is the free group monad then there's an obvious map $$S : X \times T(\{a,b\}) \rightarrow T(X \times \{a,b\})$$ which I guess is the left strength.
For example, $$S(x,\,a^2b^{-3}a) = (x,a)^2(x,b)^{-3}(x,a).$$
Indeed, this same definitional pattern can be simulated for any monad on $\mathbf{Set}$ that I can think of.
Question. What's an example of a monad on the category of sets that is not a strong monad in a natural way?