Nobody mails me anything for any other holiday but on Purim my relatives send hamantaschen in the mail! What's the origin of this tradition?
Welcome to Mi Yodea Sam! It could be that your family sends Hamantashen to you on Purim as part of the Mishshloach Manos - food sent to each other in order to cultivate camaraderie and spark/deepen relationships. It's a unique mitzvah on Purim to send food to one another as Esther says in the Megilla (9:22)
כַּיָּמִים אֲשֶׁר־נָחוּ בָהֶם הַיְּהוּדִים מֵאוֹיְבֵיהֶם וְהַחֹדֶשׁ אֲשֶׁר נֶהְפַּךְ לָהֶם מִיָּגוֹן לְשִׂמְחָה וּמֵאֵבֶל לְיוֹם טוֹב לַעֲשׂוֹת אוֹתָם יְמֵי מִשְׁתֶּה וְשִׂמְחָה וּמִשְׁלוֹחַ מָנוֹת אִישׁ לְרֵעֵהוּ וּמַתָּנוֹת לָאֶבְיוֹנִים׃ - the same days on which the Jews enjoyed relief from their foes and the same month which had been transformed for them from one of grief and mourning to one of festive joy. They were to observe them as days of feasting and merrymaking, and as an occasion for sending gifts to one another and presents to the poor.
Although it may be nice to send matzah before Pesach or apples with honey for Rosh Hashana, there is specifically a mitzvah on Purim to give Shalach Manos - literally "send food"