Once again, it's time to take inspiration from @theevilanonblog's post about Frankenstein and look at different possible interpretations of another work of fiction.
In this case, it's the Toy Story series.
The Toy Story films are about parenthood, with the toys as parental figures to the children who own them. In the first film, Woody is like a father dealing with the fact that his son, who used to idolize him, now has another, "cooler" male role model (e.g. a teacher, a coach, a celebrity hero, or even a stepfather), but ultimately realizing that this doesn't diminish his son's love or need for him. Then the second film is about accepting that someday your child will grow up and leave the nest, so you should treasure every moment with them while you still can. The third film is about realizing your child is all grown up and letting them go, while the fourth film is about coping with "empty nest syndrome" and finding a new purpose outside of being a parent, and the fifth film is about feeling irrelevant in your old age, but ultimately realizing that you still matter and that your impact on your children will last for the rest of their lives.
The Toy Story films are about growing up, with the toys as symbolic children and the children as their parental figures. In the first film, Woody's jealousy of Buzz is like a child's jealousy of a new baby sibling, with whom he ultimately bonds and learns to share his parent's love. The second film reflects the childhood realization that someday you'll grow up and have to leave your parents' care, then the third film is about finally reaching adulthood and leaving the nest, and the fourth film is about finding your purpose as an adult, with the message that you shouldn't be tied down by "obligations" to your family or by a sense of what your life "should" be like. I'm not sure if the fifth film still reflects this theme, though.
The Toy Story films are about friendship. The series' theme song is "You've Got a Friend in Me," after all. In the first film, Woody's jealousy of Buzz mirrors the jealousy that a child often feels when their best friend finds a new friend, followed by realizing that it doesn't need to be a competition, but that all three of them can be friends. Then the second film's theme is that friendship is worthwhile even though it means risking hurt and loss, while the third film is about saying goodbye to a friend and opening your heart to new friends. Some fans argue that the fourth film loses this theme by having Woody leave his friends for romantic love in the end. But others argue that it does maintain the theme, showing that some attempted friendships just don't work out (Woody and Bonnie) and that sometimes the best thing to do for a friend is to wish them well and let them go (as Buzz and the others do for Woody). Last but not least, the fifth film is a reminder of the importance of real, human connections in a world increasingly connected by technology.
The Toy Story films are about relationships. All types of relationships: friendship, romance, parent-child, etc., all embodied in the toys' relationship with their child owners and with each other. Everything covered in the "toys as parents," "toys as children," and "toys and their owners as friends" analyses can be covered in one interpretation this way: it's about all forms of love.
The Toy Story films are about accepting change. The world of toys is even more subject to constant change than the human world: children adding new toys to their collections, losing interest in some toys in favor of others, and eventually outgrowing their toys, which go on to new owners, or else become ownerless. So a film series about toys is the perfect vehicle to explore this subject. Each film has the toys facing changes they don't want, but which they ultimately make peace with. And each film ends with a permanent change to the status quo: in the first film, Woody and Buzz living happily side by side, in the second, the addition of Jessie and Bullseye to Andy's toys (and the quiet understanding that someday Andy will outgrow them all, but not just yet), in the third, the toys being passed down from Andy to Bonnie, in the fourth, Woody leaving to start a new life with Bo Peep, and in the fifth, the inclusion of Lilypad in the toys' family and Bonnie's new friendship with Blaze.
The Toy Story films are about fulfilling your lifeās purpose. The toys have a clear sense of what their purpose is: to make a child happy. But each film raises different questions about that purpose. Can you still fulfill it when your owner loses interest in you? Do new toys or other things that make them happy make you obsolete? What about when they grow up and don't need you anymore? Is your purpose to make one specific child happy, or can you move on to another child? Can you have a purpose other than making children happy? Or can you find ways to make children happy without being played with? Meanwhile, Buzz's journey in the first film of accepting that he's a toy also deals with this theme: he thought he was a space ranger whose life's purpose was to fight evil, and he feels lost and broken when he loses that identity, but in the end he finds new purpose in making Andy happy along with the other toys.
The Toy Story films are just about what it would be like if toys were secretly alive and cared about their child owners. They don't need to be an allegory for anything else.