Where is the error in the following?
In the category of all posets with increasing maps:
With product order it becomes a monoidal category.
With "product" which maps a pair posets into the set of Galois connections between them it also becomes a monoidal category.
So there are two ways to define the tensor product. Thus product order between two posets is isomorphic to the set of Galois connections between two posets.
As Mariano Suárez pointed out in the comments above you can have many non equivalent monoidal structures over a category.
As an example consider $\mathbf{Set}$: both the cartesian product and the coproduct induce a monoidal structure over $\mathbf{Set}$, nonetheless they are far from being the same thing. If you take the set $[3]$, whose element are the numbers $0$, $1$ and $2$, you have that $[3]\amalg[3]$ has cardinality $6$ while $[3]\times[3]$ has cardinality $9$, hence they cannot be isomorphic. So the two monoidal structures cannot be equivalent since their tensor products fail to give even isomorphic objects for the same arguments.