What did Richter mean regarding an acid base reaction?
I am afraid you are mixing all modern and old concepts. Moles and the concept of limiting reagents did not exist in Richter's time. 


  It took 615 parts by weight of magnesia (MgO), for example, to neutralize 1000 parts by weight of sulfuric acid


This relatively famous statement has nothing to do with law of multiple proportions but it illustrates the concept of equivalents. All this observation shows is that it always took the same amount of X to react with Y. Hence the ratio of X/Y remained constant.

If we take 1000 parts of sulfuric acid as a standard, then 615 parts by weight of MgO will neutralize it. It will take 672 parts of ammonia to neutralize the same amount of acid.

Note carefully this concept does not even need the knowledge of atomic weights.

Addendum
Also note that Richter's values are incorrect as shown in the comments i.e. if we use moles. Most likely the reason was purity of his magnesia and sulfuric acid. The inaccuracy of his values was realized in the late 1860s but we should respect his idea of constant ratio of reactants. I don't know enough German to check his entire three volumes of books nor have access to them. However a table from 1860s shows the comparison of corrected values (which are still slightly off):

An Introduction to Chemical Philosophy, according to the Modern Theories, by Adolphe Wurtz, 1865.



"It will be noticed that for many bodies Fischer's numbers are widely different from the theoretical figures, and consequently the analyses of Richter from which
they are calculated were not nearly so accurate as those of Wenzel. The inaccuracy of these analyses, and the obscurity of a perplexed explanation, have not been much noticed in giving Richter credit and some authority to
his works."