Consider $G=\mathbb{R}\times \mathbb{R}^*$, with usual coordinate-wise group operations (i.e. $(a_1,b_1)*(a_2,b_2) = (a_1+a_2, b_1b_2)$.) Geometrically this set is nothing but the real plane minus the $X$-axis.
I would like to know whether there is any geometrical interpretation of the group structure? Does this group appear naturally (e.g. in Physics?) anywhere?
On
A problem with your statement "Geometrically this set is nothing but the real plane minus the X-axis" is that we are almost hard-wired to think that the appropriate operation on the real plane is vector addition which is not what we have here at all.
I think a more appropriate way to think of this group geometrically is to first note that $\mathbb{R}^{*}\simeq\mathbb{R}^{>0}\times\{\pm 1\}$. Then we can use the (continuous) homomorphism $\log$ to see that $\mathbb{R}^{>0}\simeq \mathbb{R}$, where the operation on the right hand side is addition.
That is, another way of looking at your group is to view it as $\mathbb{R}\times\mathbb{R}\times\{\pm 1\}$. It's a couple of planes, with a rather odd operation.
Groups arise very often in a geometrical context, e.g., as symmetry groups, or as groups of transformations of a geometrical space, in physics and mathematics. There is also a whole branch called "Geometric Group Theory". I don't know a particular interesting meaning of the direct product $\mathbb{R}\times \mathbb{R}^*$, but the semidirect product has an interesting geometrical interpretation, namely as the group of affine transformations, $${\rm Aff}(R)=\mathbb{R}\rtimes \mathbb{R}^*$$ see the affine group. Note that $GL(\mathbb{R})=\mathbb{R}^*$.