Is this a superior measure of the size of sets, to cardinality?
If we use points on the real line as an example of the number of elements in a set, we know that any segment has uncountably many points and any set of discrete points is countable.
Surely therefore, we can create a measure of the number of points in any subset of $\mathbb{R}$ as follows:
Sum the length of any segments into one measure; a real number (or uncountable infinity) $r$.
Sum the number of discrete points (including any endpoints of closed segments) into another measure; a natural number or countable infinity $n$.
Then we will have a measure $\lvert X\rvert_m=\{n,r\}$ with $n$ a natural number or countable infinity, and $r$ a real number or uncountable infinity.
This measure has the advantage over cardinality that it can order the sizes of some sets which are in-between some uncountably infinite sets and others. For example, by this measure the open and closed line segments from $0$ to $1$ have measures:
$\lvert[0,1]\rvert_m=\{2,\infty\}$, and
$\lvert(0,1)\rvert_m=\{0,\infty\}$
respectively. Whereas by cardinality, the larger set has the same measure as the smaller.
Do measures of this nature exist, and can this be translated to set theory? It would seem that this measure, to some degree, sidesteps the problem of the continuum hypothesis, which limits the power of cardinality.
If by "discrete point" you mean "point not contained in an interval" (note that usually "discrete point" would generally be taken to mean "isolated point") then there can be uncountably many discrete points! E.g. if we take $X$ to be the set of irrationals, then every point in $X$ is discrete in this sense, but there are continuum-many irrationals. So you need to allow $n$ to be an arbitrary cardinality $\le$ the continuum.
And this means you don't sidestep CH at all. Note, in particular, that if $X$ is an uncountable set of reals of size less than continuum then every element of $X$ is discrete in your sense, since each nontrivial interval has cardinality continuum. (This is related to your previous questions about intervals and CH; I suspect that ultimately you're conflating "point not contained in any interval in $X$" with "isolated point of $X$".) At the end of the day, for "interesting" sets of reals (that is, sets relevant to CH) counting the discrete points is exactly the same as counting the cardinality.
Finally, note that even if we expand your definition to allow $n$ to appropriately count the number of discrete points, your definition is actually less expressive than cardinality in some circumstances. E.g. consider a fat Cantor set of measure $1$ versus an interval of length $2$. Your size notion would give the first set $(0, 2^{\aleph_0})$ and the second set $(2, 0)$, so it wouldn't see that the interval is bigger than the fat Cantor set.