Why does a Fortify Restoration potion also augment seemingly unrelated skills that aren't mentioned in the name?
The answer to your question is multi-faceted but most likely due to a mixture of UI decisions.
Easily found, is the Fandom Wikia regarding Skyrim potions. If you research this page you find that potions and ingredients have up to four effects. This is also experienced in playing when eating ingredients and learning one of its effects each time.
How this may be a UI decision is the space taken to display effects beyond secondary takes much visual real-estate wherefore many players want to get the basic info and move on.
Additionally, players who are interested in deep lore to include alchemic properties will likely take their own notes (The fandom Wikia lends credit to this in that it exists from player input and collected notes.)
There are at least four other first and second page search results on potions and what happens when, why, and how within Skyrim.
Please also note this is an answer to the main title and body question, not the OP self-answered theory.
Based on answer here What is the Math Formula on Fortify Restoration Potion Improves Fortify Other Skills
The way it works is that fortify X counts as a restoration effect.
That is exactly the problem.
Also equipping fortify X is seen as applying a restoration effect that is improved by fortify restoration alchemy.
So no, it's not a bug. It's the way it's supposed to work.
I saw this post
The way I understand it, is the following.
It has something to do with removing clothes and putting them back on again.
If that's true, that explains why the PC have to put off the clothes and put it back again. Just drinking fortify restoration while wearing the clothes won't work.
That is, the effect of fortify alchemy must be added while the user is under the effect of fortify restoration.
The fortify restoration does not improves alchemy or enchanting directly. Drinking fortify restoration and then putting fortify alchemy causes it.
In other words, it's not a bug. The fortify restoration works as intended, magnifying the beneficial effect.