As per RAW - No, you can't use the spells in your spellbook as Artificer Spells

In general Artificer Spells, just like Wizard Spells or Warlock Spells are those spells that are in the list of available spells for the class.

There are exceptions to this rule. Certain class features might state that your list of available spells is bigger - for example Warlocks choose a patron that grants them certain spells that are not in the list of Warlock Spells that you would find at the end of the PHB. But: This feature states that these spells count as Warlock Spells for you.

The Wizard's spellbook for example allows you to transcribe Wizard Spells into the spellbook.

As such you can normally not infuse spells that are in your spellbook. You have to learn them through a class feature of your Artificer class that explicitly says that the spell counts as an Artificer Spell. Otherwise you would learn Wizard Spells when leveling your Wizard side, Artificer Spells when leveling your Artificer side and you could transcribe Wizard Spells into your spellbook - not Artificer Spells. Even if they are in both lists, they don't count for both at the same time.

Unearthed Arcana is playtesting material

Usual caveat: as you said yourself, Unearthed Arcana is not balanced for multiclassing and as such there may be problems when it comes to situations like these. Test the playtesting material and if you think something is too powerful/not powerful enough you should discuss with your DM how to adjust it.

Saying that every spell in the spellbook that is also in the list of Artificer Spells counts as an Artificer Spell for the purpose of class features from the Artificer when multiclassing is a possible solution. Or saying that you can transcribe spells for all classes where you have at least one level, so that you could treat the spell as an Artificer Spell.

But here you are in homebrew territory and this might have unintended side effects.

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.