Sorry for answering questions that Arturo should, but he's not there right now and I actually still support a timeline that he feels forced not to anymore, of which this trident issue is a plot point, so I may be qualified as well

Ganon is a pig. Phantom Ganon is not.
So? Ganondorf is well able to lend his weapon to his servant, seeing as he already lends his horse to him.
Phantom Ganon's trident bears not the same shape as the Ganon trident,
The shape is almost the same in OoT and FSA, like Arturo showed.
nor does it do the same thing (transform him into a pig).
There is actually a simple explanation to this: the trident first gained the transforming ability when it absorbed the essence of the dead King of Darkness, Ganon the pig. Many say this is never stated like that in FSA, but even if it should be fanfic, it goes very well to explain why it transforms a Gerudo man into a demon that has been alive before.
And anyway, the pig demon form is by no means the compulsory appearance that the King of Darkness/Evil King always takes, as seen in OoT, TWW and TP, where he can change from human to demon and back at will. It makes sense to assume Ganon has this changing ability in every single game where he appears in pig form only, besides Oracles where he never was human to begin with.
Tell me why they should be the same.
The reason you brought up the trident issue was because it apparently hinders a smooth transition directly from OoT to ALttP, a segment that for example my timeline has like that. But, saying OoT's trident is the same as in all other games will conveniently negate this particular hindrance, so that's why in my opinion they should be the same. Personal preference. I won't deny that stating that serves my timeline, but on the other hand my placement is not *built* on this as some fact. It's just a vague attachment point for a tiny bit of fanfic to better connect OoT > ALttP > OoX+LA > FSA.
FSA established that the Trident is tied to the Ganon pig form; I think that makes it clear that it's intended to be associated with pig Ganon, not pigless Ganondorf.
In OoT, Ganondorf used the Triforce of Power to become a pig, and he carried two huge swords in the final fight that may or may not be a transfigured form of the trident. This is extreme fanfiction, but as far as I can think of, assuming such a thing doesn't conflict with anything. If people can assume a new Hyrule is discovered after TWW, I may as well assume the swords are a magically transformed trident

Anyway, the original way to become a pig was the ToP. And this demon form may have been housed inside the trident after Ganon's demise in one game (depending on your placement, several are candidates for that, like ALttP or Oracles). That's why Zelda in FSA calls Ganon the ancient demon reborn, clearly saying that the *demon form* was reborn, not the Gerudo form (although that one was reincarnated for sure, too, but Zelda doesn't have to know it).
The trident, by the time of FSA, grants the power to become the King of Darkness, something which the ToP never was said to do, although it effectively did. Again, the title King of Darkness does not necessarily imply that Ganon has to take on pig form. Just because he is in FSA, does not mean that the trident forced this appearance on him.
I find it sensibly to assume the trident gives him the power to transform into a pig demon, but like the Ganon(dorf)s in OoT, TWW and TP, he should be able to switch back to human whenever he likes. It just wouldn't have benefitted him in the boss fight if he had done so, and it spared the programmers effort to create his Gerudo sprite (which I'm personally glad for, because this way allows to make one's own picture of what he looked like).