Reactivity of Ammonium
Let's compare ammonium cation NH4+ and the neutral molecule, NH4 with other metals. 

As Ed V pointed out, with a mercury cathode and low concentration of H+, NH4+ can be reduced to NH4 electrolytically (also with sodium amalgam), which forms an amalgam that is stable for a short period of time: the mercury bubbles up, then collapses, evolving NH3 and H2. Ammonium amalgam was discovered in 1808 by Humphry Davy and Jöns Jakob Berzelius. 
https://en.wikipedia.org/wiki/Amalgam_(chemistry)

A recent (2016) paper http://scripts.iucr.org/cgi-bin/paper?xk5030 gives the crystal ionic radius of ammonium as falling between 140 and 154 pm, depending on the anion; this compares nicely with the atomic radius of mercury which is 150 - 155 pm, so an alloy is not inconceivable. 

Metallic ammonium (NH4) has been considered for some off-world chemistry in a 1954 paper. The phase transition of mixed crystals of hydrogen and ammonia to metallic ammonium was found (by calculation) to take place at pressures almost certainly less than 250 kilobars, suggesting that Neptune and Uranus contain large deposits of metallic ammonium.

http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?db_key=AST&amp;bibcode=1954MNRAS.114..172B&amp;letter=0&amp;classic=YES&amp;defaultprint=YES&amp;whole_paper=YES&amp;page=172&amp;epage=172&amp;send=Send+PDF&amp;filetype=.pdf

But on our world, the reaction of a metal like iron with a high concentration of NH4+ cation would produce H2, not NH4. The reaction could be examined on the basis of the pH, or concentration of H+ ions. As it turns out, the potentials are simply related to the concentration of H+ ion, which depends on the anion.

A 0.1M ammonia solution (anion = OH-) has a pH of 11.122 (http://www.chem.science.unideb.hu/Solutionsmonobasic.pdf). A solution of NH4Cl (anion = Cl-) gives a pH of 5.13 (https://socratic.org/questions/calculate-the-ph-of-the-following-aqueous-solutions).

When we displace from a solution of ammonium salts, we wind up displacing hydrogen. You could imagine that adding iron or zinc to a solution of NH4+Cl- could evolve neutral NH4 which immediately decomposes to NH3 plus a hydrogen radical H., which then dimerizes to H2, and the NH3 immediately redissolves in the water. Far better to think of the NH4+ dissociating in the aqueous solution before reacting with the iron or zinc, in other words, the reaction with the metal is a reaction of H+ (or, actually, hydrated H+, or hydronium ion) after dissociation of NH4+ in the presence of H2O.