No, non-isomorphic group structures on the same topology of the real line are not possible. For the sake of clarity I’ll refer to the topological space as to the real line, not ℝ, to emphasize we don’t rely on a known arithmetical structure, only on the known topological structure, and have some (unknown) group structure.
First of all, let’s define the family of left group operations (currying):
Gx(y) = x • y .
For each x: Gx must be a homeomorphism of the real line. Such homeomorphism may be, in principle, either order-preserving or order-reversing. Since for the identity element e: Ge must be the identity map, Gx depends on x continuously, and the real line is connected, for any x: Gx is order-preserving.
Let’s go further. Except for Ge, any Gx may not have fixed points. It is easy to check that when x > e it shifts the line to positive side and for x < e it shifts the line to negative side. We assume that inherited certain total order structure from ℝ, but it is generally not important which of two possible order structures on the real line are we using.
From this follows that our group is a totally ordered group that implies that is torsion-free.
Now you can think of e as if 0 and of an arbitrary distinct element as of 1, and tedious 19-century mathematical analysis-fashioned reasonings can demonstrate that we haven’t anything but addition of real numbers up to isomorphism.