If you copy a unique spell or formula, is it still unique? How many people have to have access before it is simply rare? If it is distributed through a society network or other organization does it become uncommon?
This is described as an intrinsic trait, but that doesn't make sense...
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Rarity tags are a mechanical thing for understanding a game’s base state before a game starts. GMs are free to adjust rarity tags for anything at any time:before a campaign, during, or even after (for the next campaign). It kinda has to be free form because GMs might not want a unique item to become rare/uncommon/common just because the players are fixating on it for their own characters.
Checking Archives of Nethys for unique spells, I see that all five unique spells are Impossible spells from Impossible Magic. The rules for Impossible spells say:
Casting Impossible Spells
Impossible spells are counted separately from a caster's list of learned spells, spell repertoire, or similar spellcasting class features. They are in a separate list of their own and don't consume a spell slot when cast. While most impossible spells are simply recorded as knowledge within a book, it's possible for them to exist in the form of a scroll, wand, staff, or other magical implement. Even in such forms, the power required to cast an impossible spell means only trained spellcasters can use the item. The items are also always consumed in the process of casting the impossible spell, preventing it from becoming an artifact or relic that can be passed on easily.
According to those rules, casting an Impossible spell from a wand or staff destroys the item. I can imagine that copying an Impossible spell from a spellbook would cause the page in the original spellbook to go blank. The Impossible spell would maintain its unique rarity by destroying its previous record.
If you copy a unique spell or formula, is it still unique?
Who cares?
Rarity tag is an indication. There's no rule element attached to it. So if you want to degrade the rarity tag to Rare, do it. It won't have any effect anyway.
If you copy a unique spell or formula, is it still unique? How many people have to have access before it is simply rare? If it is distributed through a society network or other organization does it become uncommon?
This is described as an intrinsic trait, but that doesn't make sense...
I'm not sure how this could happen. The only unique spells are the Impossible Spells and they're surrounded in restrictions that make it very difficult for them to become more widely known. More importantly: their campaign function is a singular thing that you quest for as part of a story or specific goal.
If a GM did decide to allow it to become more widely available? Then yeah, the rarity tag could be changed. All that really does is change the DC to identify or recall on it.
For items... a unique item probably doesn't have a formula to copy. Unique items are typically either artifacts or other singular things that there's no straightforward way to make.
If you do manage to reverse engineer it and get a formula? Then yeah, maybe that changes the rarity. But that's also subject to the GM allowing you to do that.
As rarity is a tool for the GM in the first place, the GM having things happen that change the rarity is perfectly fine. You may be putting too much stock in rarity. Fundamentally it's a tool for GMs to use to limit availability by making things that aren't common in Golarian marked as such out of the box. GMs can use that or ignore it as they wish. It doesn't do much else.
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For items yeah the actions of players could change the rarity, Rarity has changed on items that were adventure specific and original and experimental creations of the characters within those adventures which were later reprinted as generic rulebooks like the Treasure Vault. So an alchemist in an early adventure absolutely did manage to make their creation widespread to the point that its not common across golarion.
Unique items are pretty much artifacts and relics as Tridus mentioned. But it can also be items that only exist due to extreme circumstance. Or just that they are unique creations that have not been repeated or shared canonically either due to the difficulty in crafting them or needing some rather extreme requirements. Or just that the creator is unwilling to share his creation.
I would not say Common across Golarion, as the Rarity in the books is very much specific to the Inner Sea Region.