"An un-birthday present" is a gift given on a day that isn't your actual birthday. Alice questions whether she prefers birthday presents, showing her fondness for celebrating her special day. The concept highlights the playful and whimsical nature of the conversation.
Humpty Dumpty counters by challenging Alice's understanding of time, asking about the number of days in a year, emphasizing the difference between special occasions and ordinary days. Their dialogue reflects the absurd and humorous tone typical of Lewis Carroll's work.