Actually they do. The zodiac sign constellation of a day correspond to the constellation over which the sun while be at noon. So you do have a zodiac sign constellation associated to your day of bird.
But depending on how your culture group stars in constellations you might be from a different zodiac.
If you want to follow a classical westerner map of the sky, they will be 13 of them.
Also, note that the zodiac signs won't cut the year in equal parts. Some constellations are just bigger than other and the sun path across the sky changes more or less quickly from a day to another depending on the time of the year.
Yeah, I also didn't like the wording of the meme. It's like saying "Santa Claus doesn't exist". He does exist, he's just not a real person, but he exists as a concept and influences millions of people.
I'm aware that he's the basis of the modern Santa Claus (I've actually been to Demre), but I would argue that the modern Claus is a separate entity/concept at this point.
I would go even further : Santa Claus is presented as a actual person that we all know does not exist even though the concept exists. But noone ever pretended that Zodiac are a physical thing, it's a concept and concepts can't get more real that being conceptualized.
Check out wikipedia. There is two pages each time one for the zodiac sign and one for the constellation.
And if you go to the french version, the dates at which the sun cross the constellation are each time in the introduction. It's the date of the first paragraph. The ones the second paragraph is the dates is in astrology, for disambiguation. The comparaison are harder with the english articles.