Her love letters--to and from Daddy--were in an old box,

Her love letters--to and from Daddy--were in an old box, tied with ribbons and stiff, rigid-with-age leather thongs:1918 through 1920;... Why (Daddy) in this sentence was written with a capital D?

The rule of thumb is that if you can replace the name Daddy/Mommy/Dad/Mom with a name, then it gets capitalized.

I received 20 letters from (Daddy.) I received 20 letters from (John.)

That works, so capitalize Daddy.

I received 20 letters from my (Daddy.) I received 20 letters from my (John). Wrong. Using my dad means no capital, because one should not say my John normally.

Guide: capitalize words such as "Mother," "Father," "Grandmother," "Grandfather," "Son," "Daughter," and "Sis" when they are used in place of the person's name. Do not capitalize them when they follow possessive pronouns such as her, his, my, our, your.

Mother wrote to Father every day. Mary wrote to John every day. - correct

My Mother wrote to Father every day. My Mary wrote to John every day. - incorrect. My Mary is not used this way.
My mother wrote to Father every day. - correct