First of all let me say that I find it difficult to articulate exactly what it is I want to ask, but I will try to be as precise as I can.
I'll start by clarifying I'm not trying to ask about what cardinality is, or why it is useful/interesting. What is bothering me is that when talking about sets, the words "cardinality" and "size" are treated as synonyms, and I've realized I'm not sure why that is a good idea, especially when concerning students new to the concept of cardinality.
To help you understand where I'm coming from, when I first learned about cardinality I struggled with it a lot. It just didn't make sense. Everything seemed rife with paradoxes and nonsensical results. In retrospect I realize that the moment I began to become comfortable with cardinality was the moment I abandoned any association I had of it to what my mind thought "size" should mean and instead began looking at it as simply being about bijections which obviously acted like bijections should.
To give an example, before I was educated about cardinalities, if I was asked how many even numbers there are compared to natural numbers, or positives compared to reals, I'd probably answer "half as much", and I suspect that's not uncommon.
Obviously that answer is wrong from the perspective of cardinalities, but isn't that evidence that cardinality just doesn't behave like something that fits our common ideas of what a "size" is? Perhaps we could come up with a different notion of size where those 'intuitive' answers do make sense?
Personally, I feel like when I use the word size I think of concepts like distance, mass, or volume. Cardinality seems like it fits the notion of "isomorphism of sets" more than anything else.
So to sum up the question: is there a good reason why we'd want to actively associate the notion of "cardinality" primarily with the word/concept of "size"? Or is it perhaps a mostly historical remnant that does more pedagogic harm than good?