There are two levels of meaning here. First, what does "day" mean in the context of the story, and second, what is the meaning of the story itself.
Suppose I asked, "What does the word 'trees' mean in the following passage?"
Judges 9:8-15 NRSV
The trees once went out to anoint a king over themselves. So they said to the olive tree, "Reign over us.' The olive tree answered them, "Shall I stop producing my rich oil by which gods and mortals are honored, and go to sway over the trees?' Then the trees said to the fig tree, "You come and reign over us.' But the fig tree answered them, "Shall I stop producing my sweetness and my delicious fruit, and go to sway over the trees?' Then the trees said to the vine, "You come and reign over us.' But the vine said to them, "Shall I stop producing my wine that cheers gods and mortals, and go to sway over the trees?' So all the trees said to the bramble, "You come and reign over us.' And the bramble said to the trees, "If in good faith you are anointing me king over you, then come and take refuge in my shade; but if not, let fire come out of the bramble and devour the cedars of Lebanon.'
It seems clear to me that the word "trees", within the context of the story, is not used allegorically. It refers to perennial woody plants. But that's not the main point. The whole story is an allegory. There are clues to this: The trees can talk; they desire a king.
Likewise, the word "day" in the context of the story of Genesis 1 refers to "evening and morning", not an unspecified time period. But what is the purpose of the story itself? That's the larger meaning that must be considered.
Like the story of the trees in Judges 9, Genesis 1 gives us clues that the story itself is not meant as history. For example, the first three days pass without the sun. Most Christians even in ancient times caught that hint.
But if the story is not meant as history, then what does it mean? I think dleyva3's answer, ironically, points us to the non-historical meaning. Genesis 1 is a liturgical story. Its purpose is to remind us, as we gather for worship every seventh day, that we are made in the image of God. The story reinforces that, not only by stating it outright, but by incorporating the sabbath day into the creation story itself.