Non-isomorphic Lie groups which are isomorphic as topological groups

Question

The resolution to Hilbert's fifth problem entails that any topological group which happens to have the topology of a manifold can be given a differential structure which makes it into a Lie group. My question is, is it possible that there are two distinct ways of doing so? To put it another way, as the title says, can two Lie groups be non-isomorphic considering the differential structure, but isomorphic as topological groups?

Answer 1

The resolution the Hilbert's fifth problem entails that any topological group can be given a differential structure which makes it into a Lie group.

No, that's not even close to be true. The fifth problem additionally assumes that the topological group is a topological manifold under the hood. There are lots of topological groups that are not manifolds.

can two Lie groups be non-isomorphic considering the differential structure, but isomorphic as topological groups?

No. It is well known that every continuous homomorphism between Lie groups is automatically smooth.