These typologies can easily confuse as much as they explain.
For example, binary data, as introduced in many introductory texts or courses, certainly sound qualitative: yes or no, survived or died, present or absent, male or female, whatever. But score the two possibilities 1 or 0 and everything is then perfectly quantitative. Such scoring is the basis of all sorts of analyses: the proportion female is just the average of several 0s for males and 1s for females. If I encounter 7 females and 3 males, I can just average 1, 1, 1, 1, 1, 1, 1, 0, 0, 0 to get the proportion 0.7. With binary responses, you have a wide open road then to logit and probit regression, and so forth, which focus on variation in the proportion, fraction or probability survived, or something similar, with whatever else controls or influences it. No one need get worried by the coding being arbitrary. The proportion male is just 1 minus the proportion female, and so forth.
Almost the same is true when nominal or ordinal data are being considered, as any analyses of such data hinge on first counting how many fall into each category and then you can be as quantitative as you like. Pie charts and bar charts, as first encountered in early years, show that, so it is puzzling how many accounts miss this in explanations.
Put another way, you can classify raw or original data as first reported and as appearing in say the cell of a spreadsheet or database. But its original form is not immutable. Imagine something stark like a death from puzzlement from reading too many superficial textbooks. That can be written on a certificate, but statistical analysis never stops there. There is an aggregation to counts (how many such deaths in a area and a time period), a reduction to rates (how many relative to the population at risk), and so on.
So, how the data are first encoded rarely inhibits their use in other ways and transformation to other forms. The etymology of data is here revealing: translating the original Latin literally, they are as given to you, but there is no rule against converting them to many other forms.