'K day' is an abbreviation of Christmas like 'x-mas' and there have been numerous suggestions as to its origin. Some theorists suggest it reflects the Eastern Jewish tradition against inscibing an X (the sign of the cross) in legal documents.
K-day or kiloversary refers to the day when someone or something is an even multiple of 1000 days old. Such as: when someone is 35 years, 7 months and 3 days old they can celebrate their 13th k-day or kiloversary.
I used WolframAlpha to figure out what my k-day is.