I think synthetic Picard would be considered to be same legal person as human Picard. The authorities would just treat synthetic Picard as a continuation of human Picard. After all, we don't see any indication that Starfleet is treating synthetic Picard as a different person. They seem to be treating him as the same person as original human Picard.
First of all, we don't know if everyone knows.
Laris seems to know, but does everyone?
The process, creates twins.
Where is the other Picard Synth, and are they both legally Picard?
What if there were a hundred of them?
What if a new personality was put into a Picard Synth who is not Picard, but he still physically registers as Picard?
Did Shinzon have the rights to the Vineyard, and share original Picards authority to command the Enterprise?
Basically it all depends on how much headway Voyager's EMH made on establishing basic rights for artificial lifeforms.
Are we even totally sure that original Picard is dead?
Dude could be in a coma.
The timing of his "death" is suspicious.
If original Picard had died a day earlier or a day later, he would have been useless to the synth agenda to establish basic rights. Either they killed him or he's still alive.
That's if Picard wasn't already a Synth, and everything that happened from the pilot, the dreams about Data, was just a play to to trick the fleshies into handing over basic rights to the synths.