The formulas you are writing seem to arise from [Cartier duality][1], 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".  <b>G</b><sub>m</sub> 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.


  [1]: http://en.wikipedia.org/wiki/Group_scheme#Cartier_Duality