I guess I'm saying: such a basic mistake in terminology in the abstract doesn't bode well for the soundness of the paper in general. Being able to properly tell apart the terms "real number" and "non-computable number" is a fairly basic qualification.
A quick check on the author leads to https://en.wikipedia.org/wiki/Nicolas_Gisin. I think it's going to be rather safe to conclude that this author is well-aware of the difference between the computable and non-computable parts of R.
It's not a "qualification", it's just some using a slightly different set of definitions for convenience. Anyone "qualified" to read the paper understands the meaning.
To nitpick: to say that "in practice" you won't select a rational is understating the case. Being uncountable, the set of irrationals is infinitely larger. The chance of selecting a rational "randomly" is zero as far (heh) as zero has any practical meaning.
How does one go about truly randomly selecting a real number?
This sort of depends on whether you're talking about "randomly selecting a real number" in the real word or in the mathematically abstract world. In the real world, you can't do it, because almost all the real numbers are non-terminating decimals. Or to put it another way, any selection process is a computation, and almost all the real numbers are uncomputable and so can't be found by any computation.
To randomly generate a real number in the abstract, I suppose you could do a random walk of infinite steps on the real number line, with each successive step randomly going left or right and shrinking the distance walked. Off the top of my head I'm not sure if that's rigorous, but it sounds like it would work.
Simply write its binary expansion, by flipping coins, (heads=1, tails=0), forever.
Very carefully.
A computable number can be a normal number.
So while your objection is valid, in practice any truly randomly selected real number is non-computable.