I'm actually not sure how many "paid vacation days" I get. It doesn't seem particularly relevant. There are certain times in the year when I know I can go away for a month, and other times when I know I have to work, and it all works fairly well this way.
Where I teach, faculty and professional staff who work less than 10 1/2 months have no specific vacation policy. Most faculty are paid on a nine month contract. Faculty are expected to identify where and what they are working on in general for a given day of the week, basically fill out a form at the beginning of the semester. There are fixed holidays and closures at winter and spring break, which are paid. The three month summer break is unpaid, though some teach summer classes or have research grants to provide income, but vacation does not accrue then either.
Personally, I just plan my vacations for the academic breaks. It's not like one can tell a lecture hall of general education students that the professor is going to take a week vacation during the 5th week of the semester. Technically I can take sick days, but I haven't needed to. I do sometimes have conferences or academic meetings off campus, but then I plan a proctored test or get a guest lecturer for that day. I've covered other classes as a guest lecturer as well, when they needed the day.