Yes it does. In any key you have a "natural" set of chords. You are pointing out the triads but in fact you have an entire 7 note 13th chord, just the mode of that degree played in thirds.
Your formula can be extended to read.
I - I-Major Triad: 1 - 3 - 5
ii - ii-minor Triad: 2 - 4 - 6
iii - iii-minor Triad: 3 - 5 - 7
IV - IV-Major Triad: 4 - 6 - 8
V - V-Major Triad: 5 - 7 - 2 (9)
vi - vi-minor Triad: 6 - 8 - 3 (10)
vii - vii-diminished Triad: 7 - 2 (9) - 4 (11)
where the numbers 1 - 8 are the notes of the major scale, 9 - 13 are octaves of 2 - 6.
Minor keys, as pointed out in another answer, usually use the harmonic or melodic minor scale to get the leading tone to the root of the key. But the formula works if you just consider the degree numbers 1 - 8 to correspond to the notes of the scale you choose to use. For example, in harmonic minor the chords will be as follows. I will do it in A to be more concrete. A harmonic minor,
i - A min - (1 - 3 - 5) = (A, C, E)
ii - B dim - (2, 4, 6) = (B, D, F)
III - C aug - (3, 5, 7) = (C, E, G#) the 7th degree of harmonic minor is #
iv - D min - (4, 6, 8) = (D, F, A)
V - E mag - (5, 7, 9) = (E, G#, B)
VI - F mag - (6, 8, 10) = (F, A, C)
vii - G# dim - (7, 9, 11) = (G#, B, D)
Note that I'm treating 1 - 8 as an index set for the notes of A harmonic minor in order of appearance. So I don't say b3, or #7. In the formula for building a chord one usually uses the Major scale as the reference so a minor triad is always noted (1, b3, 5). I hope this does not cause confusion.
This formula will always get you the chords within a key but not the key of a chord sequence. D min can be in the key of A min, C maj, or D min, just to name a few. One needs many more bits of info to extract key from this.