First thing to note is that, if the pitch centre is A, you would never refer to it as being in G major (even though it shares the same notes) or E minor.
The second thing, modes are strict "scales", if you are describing something as being "in a mode" then it means it literally only has those notes. While this is common for, for example, celtic music, or music for instruments with a limited number of notes, or for a lot of medieval music, with modern music it's very rare for music to be literally restricted exactly to 1 particular 7 note scale (and that's true for most classical music too).
Talking about something as being "major" or "minor" though, is much more broad. For example you're probably aware of harmonic minor, melodic minor, and natural minor. These are all slightly different ( A B C D E F(#) G(#) ), so even without "breaking the rules" at all (a bad term), a song in "A minor" might contain A, B, C, D, E, F, F#, G and G#.
And, in reality these rules aren't "rules" at all (and never have been), so music in "A minor" is very likely to include other chromatic notes too, for example: chromatic passing notes (notes out of key used to fill space in melodies) are very common, as are "out of key" harmonies included for "colour" (like a G minor chord in an A minor song for example). Take a song like "little wing" by jimi hendrix for example: to call it "aeolian" would be completely wrong, because it uses other chromatic notes, but to say "it's in E minor" is perfectly correct.
***So what's the answer***??
You can say something is "in A dorian" if it very strictly only uses notes in the dorian scale. However, what's more common, is a dorian "feel", that is to say, a piece that is minor in nature but has a harmonic progression such as Am D7, Em7 G A : (i.e. G natural but F#)
In these circumstances, it's common (with jazz and pop musicians for example) to say something like it's in "A minor, with a dorian flavour" or "it's a dorian-y A minor" or "A minor with a dorian vibe" or something like that: it's not super technical, but it's a helpful way to describe something without being so restrictive as to say "A dorian" but being more descriptive than simply "A minor"
If you wanted to super technically correct, instead of saying "dorian flavoured A minor" you would say "The piece's tonality is minor, the pitch centre is A, and its melody and harmony are heavily based around the Dorian scale".