No, proficiency bonuses never stack
Your proficiency bonus can’t be added to a single die
roll or other number more than once. - D&D Basic Rules V0.1 Chapter 1, P. 7
You can only ever receive a proficiency bonus once regardless of how many sources grant you proficiency in it. When picking skill and tool proficiencies granted by a background or class, Players should try to avoid overlapping proficiencies as the benefits are nil
However a proficiency bonus may be increased or decreased based on circumstances
Occasionally, your
proficiency bonus might be modified (doubled or halved,
for example) before you apply it. If a circumstance
suggests that your proficiency bonus applies more than
once to the same roll or that it should be multiplied
more than once, you nevertheless add it only once,
multiply it only once, and halve it only once. - D&D Basic Rules V0.1 Chapter 1, P. 7
DMs can use this as a way to give a bonus to skill checks based on good roleplay and/or planning.
The Rogue's Expertise feature is an exception
Expertise
At 1st level, choose two of your skill proficiencies, or
one of your skill proficiencies and your proficiency with
thieves’ tools. Your proficiency bonus is doubled for any
ability check you make that uses either of the chosen
proficiencies.
At 6th level, you can choose two more of your
proficiencies. - D&D Basic Rules V0.1 Chapter 3, P. 27
While not allowing proficiency bonuses to be counted more than once, expertise allows a Rogue to double the benefit of their proficiency bonus to two skills (or thieves' tools) at level 1 and again at level 6.
Backgrounds granting a redundant proficiency let you pick a different proficiency instead
In the Backgrounds section of the Basic Rules & PHB, the following rule is listed:
If a character would gain the same proficiency from two different sources, he or she can choose a different proficiency of the same kind (skill or tool) instead.
(Note that, while unclear, this rule may only apply to background proficiencies.)
Custom backgrounds are also an option available to players to avoid duplicate proficiencies due to backgrounds.
The bonus proficiency is a domain feature. Domains are a class feature. The multiclassing rules state that you gain the class features when you get a new level in the new class. It then lists exceptions to this with channel divinity, extra attack, spell casting and unarmored defense. Domains are not listed, so it seems that the bonus proficiency is the more specific rule and you gain the domain proficienices.
Best Answer
You gain no skill proficiencies by multiclassing into Artificer
Per the Artificer class description on page 54 of Eberron: Rising from the Last War (or in Appendix D of the Wayfinder's Guide to Eberron), there is an optional rule addendum on how multiclassing into artificer works if you're using the multiclassing rules:
These are also reproduced on the DnDBeyond class page for the artificer. Unfortunately for your multiclass, you don't appear to gain any extra skill proficiencies by going into artificer; only the tool proficiences for thieves' and tinker's tools.
The original UA Artificer from 2017 doesn't include multiclassing rules (as NautArch notes in comments, UA classes normally don't) so if you're still working from that version, that might be why you can't find these rules. Multiclassing rules were first added in the early 2019 UA version (as a sidebar at the end of the document) and finalised with the release of E:RftLW.