SynthID is a mathematical property, based on how the neural network that generated the image "sees" images. Basically it uses knowledge of the neural net, and a password, to make the image weird in only that networks eyes. Similar idea to visual hacks, where a minor random noise pattern added to an image makes an ai think it is something entirely else.

To make an image get a synthid, you have to make an ai generate it. If a tool adds captions and gives it a synthid, then it changed the entire image and sloppified it, not jsut those captions. It's now an ai's impression of what the original image was.

I suppose this is possible to happen in practice, but I've only seen something like that with pre ai-pocalypse things. Namely I recall some ai noise filters making human voices in old videos sound ai generated. Which back then people weren't sensitive to but now sticks out cause we had to learn to detect those tells.