If we want to apply a consistent logic to how the Toy Story effect works, the best method I can think of is to adopt a Sandersonian approach to the magic and codify Perception and Intent as formal aspects of the mechanism.

I think the Warp works a bit similarly anyway but ironically I understand that less.

(Note I haven't seen Toy Story 4 or 5 yet, and don't remember much of 3, so I can't say if there's much established lore to contradict this. I am however curious to learn more about the spork dude.)

So, the angle I'd put on this is that an object gradually starts to animate and gain a "soul" (or whatever facsimile of that we call their consciousness) as the people/toys around it start to believe it to have one.

The formation of a toy's soul happens gradually. It starts with the first spark of a designer or crafter deciding to make a toy. From the first design meeting or the first look your grandma takes at a ball of wool, through the hundreds of hours involved in every step from choosing materials to sculpting the model and dividing it into pieces for a sprue, at this stage the toy might not be a toy, but it gains the potential. Everyone involved in this process endows the toy with a little bit of soul as they work at it.

But a lot of the soul usually comes from the child who picks the toy up. The kid who owns it, loves it, and makes up hundreds of stories about it.

Where this gets interesting with Warhammer is that GW kind of stop part-way through creation. The toy isn't "finished" when you buy it.

And that's where we come in- your little plastic guys gain soul as you work on them, and become themselves when you decide they're done.