An Artificer could make specialty items for others, but won't.
Clearly there isn't physical limitation as you have pointed out. Two or more Thunder Cannons produced by the same Artificer can exist at the same time, and the same is true for the Arcane Magazine and the Alchemist's Satchel.
However, Artificers only make more of these specialty items if they lose them. Selling or giving away these items is not even considered as a possibility. The implication is that an Artificer is simply unwilling to make their specialty items for others. I believe this restriction is similar to the Druid's unwillingness to wear metal armor or a Paladin's unwillingness to break his oath.
Each class has story elements mixed with its game features; the two types of design go hand-in-hand in D&D, and the story parts are stronger in some classes than in others. Druids and paladins have an especially strong dose of story in their design. If you want to depart from your class’s story, your DM has the final say on how far you can go and still be considered a member of the class. As long as you abide by your character’s proficiencies, you’re not going to break anything in the game system, but you might undermine the story and the world being created in your campaign.
So a typical artificer wants to keep his specialty item to himself, but even if others get their hands on the specialty item, it shouldn't break anything in the game system, because:
Specialty items are not useful to non-artificers.
Everyone else lacks the features and proficiencies to make good use these specialty items.
The Alchemist's Satchel, only works for those who know Alchemical Formula options, that is, it only works for Alchemist Artificers.
At 1st level, you craft an Alchemist's Satchel, a bag of reagents that you use to create a variety of concoctions. The bag and its contents are both magical, and this magic allows you to pull out exactly the right materials you need for your Alchemical Formula options, described below. After you use one of those options, the bag reclaims the materials.
The Thunder Cannon is a weapon, so it can be fired by anyone, but it is neither simple nor martial so only Gunsmith Artificers are proficient with it. Moreover, the bonus action reload means that anyone with the Extra Attack feature will outright shun the Thunder Cannon. In short, the Thunder Cannon is a really bad weapon for anyone who isn't a Gunsmith Artificer.
You are proficient with the Thunder Cannon. The firearm is a two-handed ranged weapon that deals 2d6 piercing damage. Its normal range is 150 feet, and its maximum range if 500 feet. Once fired, it must be reloaded as a bonus action.
The Arcane Magazine is only as useful as the Thunder Cannon.
Homebrew subclasses for Artificers should somehow ensure that their respective specialty items are useless or near-useless to everyone else. If they don't, then I would assume it is a poorly designed subclass.
The Hat of Wizardry says:
This antiquated, cone-shaped hat is adorned with gold crescent moons and stars. While you are wearing it, you gain the following benefits:
The text seems pretty clear to me - while you're wearing it, you can use it as a focus. Holy symbols for clerics and paladins work the same way (they only need to be worn, not held), so it is not unprecedented. It has a specific verbiage that overrides the general rules.
Furthermore, it seems relatively well balanced: It consumes a precious attunement slot and only works for wizard spells. If the character is multi-classed, they'll still need a different focus, even for classes that can use a more general arcane focus. Sure, it has a second property, but it's only once per long rest.
The Dark Shard Amulet has functionally similar text, except aimed at warlocks.
Best Answer
As you've noted, the rules don't address this specifically. When the rules don't address something, the procedure is to ask your DM for a ruling.
(Asking rpg.stackexchange for a ruling won't help: we're not your DM, so we don't have the authority to issue house rules for your game. :) )
Most DMs will probably say: "Oh, you need an arcane focus? Sure, you can use whatever, that's fine." In D&D, the question of what sort of focus you're using is roughly as important as the question of what sort of shoes you're wearing: you're probably wearing shoes, and it would be a problem if you lost them suddenly, but it's unlikely to matter in the course of most games.
Of course, unless your DM is using unusual rules for character creation, your artificer would not be able to begin play with a Mizzium Apparatus, because that's a magic item and you can't start play with magic items.