What I had in mind with that comment on the tuning system and do-re-mi mode naming was something a bit different.

In 12-tone equal temperament this is not an issue.

But suppose we're constructing a white-note scale based on C in Pythagorean intonation. Then both the intervals C to D and D to E have the ratio 9:8, so the first step in the Dorian scale will be the same as the second step in major.

In just intonation that isn't true. That interval will be 9/8 in one and 10/9 in the other. So, when you're constructing your mi-mode (Phrygian), your tonal centre "mi" could be two different pitches, and you will get microtonally different interval patterns depending on whether you see it as a rotation of major, a rotation of Dorian, or build it from scratch.