This is a really good question that could explore multiple theoretical propositions. My go-to for these kinds of instances is to think about the common notes between one key and another, and the hierarchy of the harmonic tones.
- In the hypothetical case (which totally happens in other songs) that instead of going from Dm to Fm, it goes from Dm to Ebm, that new key is the farthest and most unrelated key from the original Dm. Why? EVERY note of the scale would have to be changed.
- In the hypothetical case (even MORE common) of going from Dm to Em, there are not only a number of notes in the key that would have to be changed, but also the importance of the notes that would have to change. In this new Em, the original minor 3rd, F (super important!), would have to be raised to an F#, and the original 5th (A) would have to either resolve down to become consonant, or climb up to a B natural, which also used to be a foreign note.
However, by going from Dm to Fm:
- The new minor tonic actually used to be 3rd of the chord, AND the tonic of the relative Major!! That F has never been foreign to either D minor or F major, and has always been one of the important notes, in every context! This may probably THE most common note that makes most of the magic possible.
- The former 5th (A) becomes diminished (Ab), and it becomes the new 3rd. Though diminished harmony is often stigmatized, the sonority of this one, especially with the W-H octatonic scale (consecutive minor trichords), is extremely similar to a minor tonality. Also, if we talk about intervals, the defining consonance of the minor 3rd in a minor key is easy to identify and follow to the ear. Therefore, this new minor 3rd based off the former 3rd is pretty comfortable to hear. At least more comfortable that a whole new, unrelated key that drifts the whole harmony.