As far as I can tell, the formulas you are writing arise from the usual duality between groups of multiplicative type and (fpqc sheaves of) finitely generated abelian groups. You can find this in SGA3 volume 1. Statement (**) seems to be some form of the standard fact that homomorphisms from Z to G form a group isomorphic to G. This is a defining property of Z as a group. You could probably make some kind of generalization to monoids using the natural numbers, but it's not clear how useful it would be.

It looks like Greg Stevenson basically answered your question in a comment, but you replied with something about groups in general, even though such groups didn't make an appearance in your question until the parenthetical comment at the end. Could you make some effort to write your question in a more precise way, so we know what you want, and why you want it? The fact that it came from your notes doesn't seem to be a satisfactory explanation for the fact that it is almost impossible to understand.

The formulas you are writing seem to arise from Cartier duality, which gives a collection of antiequivalences of categories of group schemes and sheaves of various types. For the case of tori, one gets a correspondence with (fpqc sheaves of) finitely generated free abelian groups. You can find some of this in SGA3 volume 1 attached to phrases like "diagonalizable groups" and "groups of multiplicative type". G_m has a distinguished role as the dualizing object. It is Cartier dual to the constant sheaf Z. I don't think there is a replacement group B that satisfies all of the identities you want, because the Cartier dual of B won't be suitably universal in the category of (sheaves of) abelian groups.