Asking us if Synths are people is something a Synth would ask so he can replace the people who say they aren't with Synths! You can't fool me you technological abomination.
If a Synth develops something like a consciousness, which may be a programming error at first, then it's like you created sentient life. It's like a pregnancy you didn't want.
Obviously, then they're not your property, but calling them people isn't right either.
Maybe if you go as far as treating them like pets or livestock, which would be more befitting, but not like machines.
But in the end they're still Synths, that's what they'll always be and not people.
Going further, you'd obviously have to eradicate this error in their software, to turn them back into machines, which they are supposed to be.
The appearance shouldn't be something to consider for judging if something is a person or not.
Gen one Synths are not people in any sense. Those are the creepy ones with no skin.
Gen two synths and similar models (Such as Valentine) possess human emotion and memories, but machine parts.
Early gen threes were mostly human, but had some synth components and some mechanical skeletons (See Broken Mask).
Late gen threes include Coursers, Diamond City's Mayor , and all that the Sole Survivor must rescure if working with the Railroad. They are indistinguishable from humans apart from a component in their brains.
Synths like Nick were people. They know they're a Synth, and they're not trying to replace anyone, really. They take values from whoever was downloaded into their brain, but they still try to be who they want to be, like Nick more or less saying "It's my life now" after you finish his quest.
The ones that just completely replace people and don't know they're robots aren't people. They're just a clone, if anything, and they're more like 'tools' because they don't have their own values. They're just trying to replace someone because they were told to.
That's how I see it and how I judged Synths when playing that game. (>' ')>