Over a C Am progression, it is possible that the music changes mode from Ionian to Aeolian. Then, in effect, over that Am chord, the tonality has changed to A. The written or improvised melodies in the music can support that by emphasizing the A note and the third and fifth above it. Modal music can change tonality without changing the key signature. The modulation clues are weak in modal music, so of course it creates ambiguities for the listener. This is particularly true if the progressions are long, or if they linger on the same chord for many bars.
If you're writing out the chord changes in roman numerals, you have to pick a "home" which is labeled I. To do that, consider the music as a whole, and in particular how it starts and ends. If the music begins on and returns to the C Ionian mode, then you have a strong justification for defending the choice of C as the I note.
If you're analyzing music which shows strong signs of modulating, you may have to shift the identity of the I note. Or perhaps keep relating the notes to the original tonic, but show their functions within the new key, in parentheses.
For instance if music in C major music contains an A7 Dm secondary dominant, you can label these as VI7 II. But in parentheses or whatever, you can also call it (V I) to show recognition that it's functioning as small embedded change of key to Dm. The case for notating it this way is weakened if it's Am Dm rather than A7 Dm because the strong modulation signal is no longer there. Any sense of "we are in D minor, temporarily", if that is the intent, will have to be created by the music in less direct ways that are ambiguous.
Modes which do not have a leading tone do not support the V I cadence. In minor keys, we add the leading tone by altering to harmonic or melodic minor, and then we have V I.
When the V note of the relative Ionian occurs in a mode, before going to the tonic (of that mode) this is called a plagal cadence or interrupted/deceptive cadence, depending on the specific situation. However, this nomenclature is based on the assumption that the Ionian mode is the "master home" of modal music, and its V degree pervasively functions as a dominant, so that whenever it occurs, a cadence of some kind must be happening, and if it doesn't go to the I, it is somehow "false".
The term "plagal cadence" is useful in that everyone knows that it means, and therefore it communicates clearly. But in fact in music which is unambiguously in A natural minor, the G strongly pulls toward the A as a half step up, and has nothing to do with the G in C major that forms the V I cadence. It is a phenomenon in its own right.
The Ionian mode is not "home" and has not always been considered that; to believe so is historic revisionism.
Scales like Dorian and Mixolydian can be considered to exist in their own right, not simply as anagrams of the notes from the relative Ionian mode.
