In this case, could we think it means "write a letter" even though it didn't designate that it's about letter?
Yes! :) When Allie is saying "Why didn't you write me?", she is definitely referring to a letter even though she didn't specifically say it was a letter. Another example of this could be: "He wrote to me last week about his experiences at university." In this example sentence, the man wrote a letter to the subject. Usually in English, if it's by email or a text message, we will say it was by email or text message.
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