TITLE: Why is cardinality considered to capture the notion of size?
QUESTION [16 upvotes]: 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?

REPLY [4 votes]: Based on everyone's answers and comments I had reached some conclusions which I've decided to share in this answer. Firstly:
The notion of "Cardinality=Size" is not ubiquitous
As some of the comments revealed to me, it is not considered common understanding that the cardinality and size of the set are the same thing. Some here evidently do hold the perspective that size is cardinality, as did my teachers, but this perspective is not prevalent everywhere. The short answer to my title question would therefore be "Cardinality isn't actually considered to capture the notion of size".
There is more than one notion of set-size
There are, in fact, many. This is only natural considering that when we think of what 'size' means we may be drawing on different ideas, and those different ideas lead to different mathematical constructions. The confusion arises when we expect the word size to mean one thing (perhaps something we can't even properly define at the moment) but it is used to mean something else.
So really, deciding that cardinality is the best way to describe the size of the set is somewhat arbitrary. However, amongst the different notions of set-size, cardinality does have a special property.
Cardinality does not assume further structure on the set
Measures require the set be part of a measure-space. Density requires we discuss the natural numbers (or integers). Cardinality is what remains when we strip our sets of all their structure and properties and treat their elements as merely 'things'. Cardinality can therefore be defined on any set, regardless of context, and it is invariant under whatever structure we impose on the set. This is great, and makes cardinality a leading candidate to the title of the size of a set, if anything.
On the other hand, this means that cardinality ignores the structures imposed on sets, however natural they may seem, and can't distinguish between sets based on said structure. In that sense cardinality is a very 'coarse' concept. From the perspective of cardinality all intervals of $\mathbb{R}$ are the same, although they are very different as far as the Lebesgue measure is concerned.
I think my confusion stemmed from the fact that when thinking of examples of 'size' I came up with 'familiar' sets that had natural structures to them which also implied natural notions for size. Even if I couldn't define what I mean when I say 'size', it was defined implicitly by structures I am familiar with, and the sizes defined by those structures were at odds with cardinality.