There are several ways to construct (or if you prefer, define) the real numbers. The two most familiar are as Dedekind cuts in $\Bbb Q$, the ordered set of rational numbers, and as equivalence classes of Cauchy sequences of rational numbers with respect to a certain equivalence relation. If one constructs them using Dedekind cuts, then each Dedekind cut is by definition a unique real number. If one constructs them in some other way, it’s no longer the case that each real number is a Dedekind cut in $\Bbb Q$, but it is still true that there is a nice bijection between the set of Dedekind cuts in $\Bbb Q$ and the reals as constructed.
Saying that the reals are the numbers on the real number line is not a definition: in order to make it one, you’d need to define the real number line independently of the notion of real number. As it stands, you’re pretty much just saying that a real number is a member of the set of real numbers.