In Toy Story, the toys are alive. Do any non-toys display this characteristic as well? I don't believe so.
If it is true that only toys can come alive, is it ever explained exactly why toys, and only toys become alive? Why not toilets, or chairs or whatever?
In the Toy Story segment of Kingdom Hearts 3, it’s explained that some unknown force unique to that world causes toys in that world to become imbued with Hearts (a spiritual organ responsible for your ability to feel emotions, which is apparently distinct from a person’s soul), and that this imbuement is what causes the toys to come to life.
Their connection to the children who own them does seem to create those Hearts, and it seems to strengthen their Hearts, as do their connections to other toys, but it’s not responsible for the process of them becoming imbued with their Hearts in the first place and becoming animated - that's some process in that particular world that doesn't seem to occur in other worlds, barring deliberate magical intervention.
We can see this in this video, where Xehanort causes a Heartless to take over an inanimate toy, stating:
One heart's shadows fill the emptiness of another. See how they bring him to life? Like Heartless and Nobodies, they fit together.
This is evidence that the toy had had its heart removed, or had a "space" for a Heart to be placed in, since when a person's Heart is corrupted by Darkness, it becomes a Heartless, with their body and soul transforming into a being called a Nobody. Later in that video, we can see a couple of toys that have been animated by a Heartless, and since they've got the Heartless symbol appearing directly on them, we know that they're being directly possessed rather than simply piloted.
Later, in one of the climactic scenes, Xehanort states:
In this world, toys have hearts. And those hearts come from a powerful bond.
In a following scene, he states:
So, even empty puppets can be given strong hearts. ... But now I know that a Heart can be placed in a vessel of our choosing.
Based on Toy Story 4, where we see an animated toy chair named Chairol Burnett, and Bonnie creating a toy named Forky, it seems that objects that are not inherently toys can be animated, although it only seems to be objects used as toys. Other non-toys include a garden gnome in Toy Story 2, and Christmas ornaments. A common fan theory is that the ownership of a child and/or having a name written on them is what makes them animated, but that's contradicted by the toys in the toy store in Toy Story 2.
It is never explained in-universe. It's just part of the world. As Wipqozn notes in their comment, there's even a joke in the credits of Toy Story 4 about the lack of explanation. One character asks "Why am I alive?" to which another answers "I don't know," and the scene ends.
Some things are true because people have faith that they are true. The innocent and uncomplicated faith of a child is a cosmically powerful force. If enough children believe that toys can come alive and have a secret life of their own, reality doesn't stand a chance.
Because children supply the divine force that creates life in toys, children get to decide what is a toy and what is not. If a child plays with an object until he begins to feel some love for it, it may come to life.
The reason that most of the things that have a secret life are toys, is that toys are designed to attract the affection of children. But it's not unusual for a chair or a fork to attract that same affection.
