Gundams (and other Giant Mecha), or rather the general purpose mass producable versions Gundams tend to be the prototypes for, are classified as Mobile Suit Units. They get classed on their own because generally speaking one or two mobile suits are enough to be a major threat to entire divisions of regular soldiers, mechanized or otherwise, and armor divisions alike, due to their mobility over more terrains than standard mechanized armor, heavy armaments and armor, and their generally universal ability to give the Inverse Square Law the middle finger and react quickly to changing battlefield conditions.
More or less, yes, though in-setting Pilots and the standard grunt infantry are generally separate entities as far as chain of command goes due to the differences in personal kit and general lethality making pilots the universe's version of special forces units.
But Gundams have armor. Thus are they not Armored units? In fact, the defining thing about the original Gundams are that they are armored with gundamium, that's why they are called Gundams right? At least that's what Gundam Wing taught me.
I mean, The original Armor unit was the Knight, and they were bipedal, so I think it follows that Gundams are Armor as opposed to Infantry (I mean, Armor units do have infantry inside the armor, but they don't fight outside the armor, same as with Gundams).
Warning: Gundam Nerding Out incoming.
It really depends on which Gundam continuity you're talking about. In the Universal Century timeline, the OG Gundam, it's called Gundam because it's armored with "Gundarium", a "titanium alloy" only producible in Lunar Gravity, and it goes through several iterations throughout the many, Many installments in the timeline. Gundam Wing/After Colony timeline, the alloy is called "Gundanium" as you say, and is only producible in Zero-G conditions, but is otherwise the same thing. Gundam SEED/SEED Destiny/Cosmic Era, G.U.N.D.A.M. is an acronym for the name of the Mobile Suit's Opperating System: "General Unilateral Neuro-Link Dispersive Automatic Maneuver". 00-Gundam/Anno Domini, it's because the suits' power plants are called the "Gundam Nucleus (GN) Drive", producing "GN-Particles" using baryonic decay as their power source. Iron Blooded Orphans/Post Disaster, the Gundams are called that because of the specific mobile suit frame they use, and to my knowledge why they're called that isn't really elaborated on since the Gundam Frames are from the giant war that almost destroyed human civilization in the Sol System, which also destroyed a lot of their history and knowledge of how things like the Gundam Frames (specifically the absolutely horrific method used to increase performance by jacking the pilot's nervous system into the suit's OS) were manufactured (though they have developed new MS frames by studying and reverse engineering them to varrying degrees of success). And Finally, in The Witch From Mercury/Ad Stella, they're called Gundam because of the GUND Format interfaces they use, which is a technology originally developed for medical prosthesis animation but was later weaponized, then outlawed for the negative effects its weaponization had on pilots who used it due to the massively higher strain controlling an entire MS is compared to a prostehtic arm or leg.