As noted by a sibling comment, he's not rhyming the -iums, he's rhyming the antepenultimate syllable (selenium/rhenium, germanium/uranium, vanadium/radium, gallium/thallium, rubidium/iridium, samarium/barium). But the other thing he leans heavily into is alliteration. The third verse is chock full of this:
Holmium, helium, hafnium, erbium (all h, save the rhyme)
Phosphorous, francium, fluorine, terbium (all f, save rhyme)
Manganese, mercury, molybdenum, magnesium (all m)
Dysprosium, scandium, cerium, cesium (all s sounds, save the first)
Lead, praseodymium, platinum, plutonium (most p sounds)
Palladium, promethium, potassium, polonium (continue p sounds)
Tantalum, technetium, titanium, tellurium (t sounds now)
Cadmium, calcium, chromium, curium (k sounds now)
There's also the point to make that most element names are 3-4 syllables long, so the verses are structured with 4 elements per line. The extra-long praseodymium and neodymium are paired with the very short lead and nickel respectively, for example. In fact, prior to the last verse, the 4 elements per line is violated just once (last line before the intermezzo crams in "bismuth, bromine, lithium, beryllium, barium"). The last verse goes a regular 4 elements, then 3, and then launches into this:
* argon, krypton, neon, radon, xenon, zinc, rhodium
* chlorine, carbon, cobalt, copper, tungsten, tin, sodium
The first of these lines rhymes all the -ons at the end, and the second goes for alliteration instead.
Of course, there are a few failures to align all the elements by alliteration or rhyme. The ones in the last verse align instead by other means (first you have californium, fermium, berkelium--all named after the same place; then you have mendelevium, einsteinium, nobelium--all named after scientists). Now that's clever!
