Apparently an early usage was in inheritance law, in contrast with "per stirpes". Suppose A has children B and C, and B has child D, while C has children E and F. If A outlives his children but not his grandchildren, then with a per capita rule, D, E, and F each get 1/3 of the estate, but under a per stirpes rule, D gets 1/2 and E and F get 1/4 each. In that usage, the plural makes sense--one really does mean something like a division of property "by heads" as opposed to "by lineages". The phrase probably then got pulled into other contexts (like expressing the units of the denominator of a ratio) where the plural is incorrect without people really thinking about it....
I should also add that ancient Roman law on intestate succession used both of these ideas. If direct agnatic descendants survived, the estate was divided per stirpes among all agnatic descendants whose fathers (and paternal grandfathers, etc.) were dead, but if only collateral agnatic relations survived, the estate was divided per capita among the closest class of surviving agnatic relations.