Except when it isn't. Spelling is meaningful, there are reasons for such a distinction.
Any G to any A is a second or a ninth.
G to Bb is a minor third or tenth. For example G A# D is not how a G minor triad is spelled.
G minor as a tonal harmony will be:
i for key of G minor; ii for key of F major; iii for key of Eb major; iv for key of D minor; v for key of C minor; vi for key of Bb major. All of these are flat keys where Bb is a given.
OTOH: G to A# is the sixth to seventh degrees in B minor, harmonic minor where A# is the major third of its dominant harmony F#. These are not interchangeable. First of all a chord built by thirds is built by thirds and G to A# is an augmented second; note its use as 6 & 7 of B minor. 1 2 3 4 5 6 7 = G A B C D E F. B C D E F G A. etc
So again: G to A# in the OP is a "#9" which is another way to state 'augmented second' plus the octave, 'augmented ninth' is the literal interval. The convention in place shows 3 qualities of 9th as well as the 3 qualities of 7th. Beyond that, we do say something else. G to 'A double sharp', or G to B sharp; well we have G to B, or G to C; a 3rd or 10th, 4th or 11th normally and we might get into an absurdity.
We might on the other hand consider G to Cb as something meaningful, albeit rare. In a chord dim 4 is not going to supplant M3 if that's the look, augmented 3rd is not going to supplant a P4 in a normal harmony.
"Double sharp 9th', no: 3rd or 10th; double sharp 10th, no: 4th or 11th.
C E Fx, no; that's like G A# D.
IE: we may write a line preserving the alphabet/simplifying the copy, eg., descend F Eb Db Cb Bb Ab G.
An octatonic scale might be spelled G Ab Bb Cb Db D E F (vs G Ab Bb B C# D E F), we can't avoid doubling one of the letter names in an 8 tone scale (never do this w. 7 notes). As a decoration of G7 in a C key we have G7 major/minor b9 #11 13 vs G7b5 b9 #9 13.
So, for the soloist navigating that, A# and Bb are more or less the same (and an academic matter really), the spelling is not necessarily defined as it is in a 7-note scale usage. (this may seem exotic, but it's actually normal vocabulary in modern jazz)
Carnatic scale theory (melakarta), the default 'shuddha' in the system has everything as flat as can be (excepting a diminished 4th or 5th, out of an ideology) and still have 7 names. G Ab Bbb C D Eb Fb. We have no concern for chord names here. So a further illustration of the differing concerns, linear vs chords.
These are the real and known definitions aka limitations.
Define limits for yourself, no prob, not necessarily to anyone else tho.