It's not that uncommon to include "company holidays" in "paid days off". Denmark has that as well, I get 5 weeks of vacation, and 5 days of vacation/days-off. My employer is free to "spend" the 5 days for me.
Seems like in this case, how the company originally presents what you're getting matters. So, as you point out here, qualifying the # of days off w/ days that are set [that count as part of that allotment] would set expectations accordingly. Otherwise, still a good deal, but employees would feel misled if they want to use it other parts of the year.