And I'm not sure what you don't understand. Anything, anybody and anyone are their own words and refer to one or no-one, out of many, not one singularly.
Any day of the week is an idiom. And it's any OF THE WEEK.
This isn't rocket science. https://www.englishforums.com/English/AnySingularOrPluralNoun/wklp/post.htm
I'm a native English speaker and I disagree with TerribleT and anyone else who says it is incorrect to pair "any" with a singular noun. Imagine this exchange: a parent asks a child, "Do you want your red shirt or your blue shirt?" The child says, "I don't want any shirt!" In that context, it sounds perfectly natural and correct to me.
I would like to add some nuance here. In English, "I want any" or "I do not want any" should be paired with plural or collective or nebulous nouns. It is incorrect to pair "any" in this context with a singular discrete noun such as shirt. But there are many contexts where any can be paired with a singular noun, such as "any day" or "any person" in both "proper" and "colloquial" English.