How the theme of Alice in Wonderland relates to the concept of identity

The most obvious theme that can be found in Alice's Adventures in Wonderland is the theme of growing up.
Lewis Carroll adored the unprejudiced and innocent way young children approach the world. With Alice's Adventures in Wonderland, he wanted to describe how a child sees our adult world, including all of the (in the eyes of a child silly and arbitrary) rules and social etiquette we created for ourselves, as well as the ego's and bad habits we have developed during our lives.
Alice's Adventures in Wonderland represents the child's struggle to survive in the confusing world of adults. To understand our adult world, Alice has to overcome the open-mindedness that is characteristic for children.
Apparently, adults need rules to live by. But most people adhere to those rules blindly now, without asking themselves 'why'. This leads to the incomprehensible, and sometimes arbitrary behavior that Alice experiences in Wonderland.
When entering Wonderland, Alice encounters a way of living and reasoning that is quite different from her own. A Duchess who is determined to find a moral in everything. Trials that seem to be very unjust. But during the journey through Wonderland, Alice learns to understand the adult world somewhat more. In fact, she is growing up. This is also represented by her physical changes during the story, the growing and shrinking.
More and more she starts to understand the creatures that live in Wonderland. From the Cheshire Cat she learns that 'everyone is mad here'. She learns to cope with the crazy Wonderland rules, and during the story she gets better in managing the situation. She tells the Queen of Hearts that her order is 'nonsense' and prevents her own beheading. In the end Alice has adapted and lost most of her vivid imagination that comes with childhood. She realizes what the creatures in Wonderland really are 'nothing but a pack of cards'. At this point, she has matured too much to stay in Wonderland, the world of the children, and wakes up into the 'real' world, the world of adults.

I think that related to the theme of 'growing up', is the motif of 'identity'.
In Wonderland, Alice struggles with the importance and instability of personal identity. She is constantly ordered to identify herself by the creatures she meets, but she herself has doubts about her identity as well.
After falling through the Rabbit hole, Alice tests her knowledge to determine whether she has become another girl. Later on, the White Rabbit mistakes her for his maid Mary Ann. When the Caterpillar asks her who she is, she is unable to answer, as she feels that she has changed several times since that morning.
Among other things, this doubt about her identity is nourished by her physical appearance. Alice grows and shrinks several times, which she finds "very confusing". The Pigeon mistakes her for a serpent, not only because she admits eating eggs, but also because of her long neck. The Cheshire Cat questions another aspect of Alice's identity. He is not questioning her name or species, he is questioning her sanity. As she has entered Wonderland, she must be mad, he states.
However, it is not only Alice's identity that is unstable. Some creatures in Wonderland have unstable identities as well. For example, the Duchess' baby turns into a pig and the members of the jury have to write down their names, or they will forget them.