I'm specifically looking at the case of a character that starts as an Armorer Artificer, then multiclasses into Wizard for the majority of their career. As I read it, as long as the Artificer has applied an Infusion to their armor, they have a spellcasting focus for both Wizard and Artificer spells in the same hand they use to enact the somatic component of a spell at all times, since their Arcane Armor has covered their entire body.
I am not counting on or involving the Armorer's Arcane Armor feature in any way here, because that clearly states that the armor can only be used for Artificer spells:
You can use the arcane armor as a spellcasting focus for your artificer spells.
I am instead counting on the specific wording of the "Tools Required" subsection of the Artificer's base Spellcasting feature, which they receive at Artificer level 1:
You produce your artificer spell effects through your tools. You must have a spellcasting focus - specifically thieves' tools or some kind of artisan's tool - in hand when you cast any spell with this Spellcasting feature (meaning the spell has an "M" component when you cast it). You must be proficient with the tool to use it in this way. See the equipment chapter in the Player's Handbook for descriptions of these tools. After you gain the Infuse Item feature at 2nd level, you can also use any item bearing one of your infusions as a spellcasting focus.
Note that it only mentions that you can use it as a spellcasting focus, not as a spellcasting focus for your Artificer spells, or with any other specific prohibitions. For precedent, here is the wording on Ruby of the War Mage, which allows Wizards to use weapons as spellcasting foci for their spells:
Etched with eldritch runes, this 1-inch-diameter ruby allows you to use a simple or martial weapon as a spellcasting focus for your spells.
I'm also thinking that the specific wording of the Spellcasting Focus section of the Wizard's Spellcasting feature gives additional precedent to my reading of the Tools Required feature from the Artificer. Here's what the Wizard Spellcasting Focus section says:
You can use an arcane focus as a spellcasting focus for your wizard spells.
This tells me that there are a few options, but I'm trying to opt for the easiest available. I could get an arcane focus of some kind to cast Wizard spells, and I could use my armor to cast Artificer spells, but all of that sounds like a lot of unnecessary work, and juggling of foci, especially when I want to punch a bad guy with both gauntlets.
Am I missing anything?