Thank you for providing a link the original Stackoverflow question. I think that there is a mathematical aspect to the answer that should be examined. There is also a numerical methods aspects that needs to be considered (that is, how computers represent numbers).
If the question were "how many numbers may be represented exactly using a floating point representation," then the answer would be similar to your (2^64 - 2^53), or something like 1.843 x 10^19. Yes, Big number. I agree, too big for "several."
However, the context of the original question is using cosine. While the range of cosine can be [-infinity, infinity], the domain can only be [-1.0, 1.0]. As you probably know, cosine is only different from [0, 2*pi] and then starts to repeat itself.
So what if we framed the question as these two questions:
How many floating point numbers may be represented between 0 and 2*pi?
And how many cosines of those floating point numbers may be represented exactly by a floating point number?
A detailed analysis of this question would probably need to be migrated back to Stackoverflow. I am going to hazard a guess that there are only 5. (Can you think of any cosine values other than 0, .5, 1.0, -.5, and -1.0 that may be exactly represented in floating point? Those correspond to pi/2, pi/3, 0, 2*pi/3, and pi. I may be wrong about only 5 numbers for cosine that may be exactly represented, so the question is not rhetorical.)
The remaining cosines are probably irrational and do not have an exact FP representation.
If the original trigonometry answer meant "zero is one of 5 values that can be represented exactly," then I think the answerer used several correctly. If the actual number is 42, then several is less right. But we are not talking about a quintillion of numbers. The range and the domain are too restricted.