Examples of non-isomorphic fields with isomorphic group of units and additive group structure
An example can be the following: $\mathbb Q(i\sqrt 2)$ and $\mathbb Q(i\sqrt 7)$.
It's well known that these fields are not isomorphic: quadratic fields are isomorphic if and only if they are equal.
It is obvious that their additive groups are isomorphic.
With respect to the multiplicative groups, note that both are isomorphic to $\{\pm1\}\times\mathbb Z^{(\mathbb N)}$ (the last denotes a countable direct sum of copies of the group of integers).