A gunslinger with Spellshot Dedication has access to Basic Wizard Spellcasting and Expert Wizard Spellcasting, but not Master Wizard Spellcasting.
Yeah, I think that is a bug, not a feature.
This counts as the wizard archetype for the benefits of Basic Wizard Spellcasting.
That line doesn't mention Expert Wizard Spellcasting - which also has Wizard Dedication as a prerequisite.
So something is in error here. Either Expert Wizard Spellcasting can't be taken with only Spellshot Dedication and shouldn't be listed as an Additional Feat - or both Expert and Master Wizard Spellcasting should be included in the override allowing them to be taken with Spellshot Dedication and both feats should be listed as Additional Feats.
So I would rule that you don't need to take Wizard Dedication separately. Spellshot Dedication still counts for the Wizard Dedication needed to take Master Wizard Spellcasting.
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That said - yes, you could also take Wizard Dedication (which would give you two more cantrip slots). I'm not sure on taking the full suite of Wizard Spellcasting feats to get an additional pile of Wizard spell slots. I think that since they are Additional Feats that they are effectively the same feat that can't be taken a second time. Which is a bit strange since you could definitely take Witch Dedication and Witch Spellcasting and definitely would get an additional pile of Witch spell slots. No name collision there.
Whether the spellbooks are the same item or not is not so important. You could automatically 'learn' your arcane spells that you already know from Spellshot into your Wizard spellbook. I don't see any reason to keep those as separate items.
I wonder if the the remastered spellshot class archetype not gaining a Master spellcasting feat is an intentional limitation to prevent it from "outshining" the eldritch archer archetype. TBF, the spellshot gains earlier access to Spell-Woven Shot (instead of Eldritch Shot), Fulminating Shot (instead of Enchanting Shot; and the spellshot gets to choose the damage type, plus increases damage automatically when gaining levels instead of tying it to weapon runes), and Phase Bullet (instead of Incorporeal Shot).
Personally, I'm a fan of taking the witch multiclassed archetype with the spellshot (even with the delayed access from needing to take two other archetype feats first) for access to the primal spell list and Ceremonial Knife (as a bayonet attached to the gun).