I know this idea from Arnold Schoenberg. In Fundamentals of Musical Composition - chapter XII - he says "watch the bass line." The harmonic style of the book is Common Practice.
The figured bass method of the early part of that era associated certain harmonizations with specific scale degrees. You can loosely translate that figured bass view into functional harmony ideas. Common figured bass harmonizations used only the tonic, dominant, and subdominant harmonies. For example, the
iii scale degree, the mediant in the bass, would be harmonized with a first inversion chord, a chord of the sixth. You can extend that idea to then say degree
iii in the bass has a strong tonic association.
Let's look at another example but using the leading tone...
vii scale degree, the leading tone, has a strong association with dominant harmony and the standard harmonization in figured bass would be a first inversion chord
V6. If we add a third voice to create a complete triad there are two choices: a first inversion chord or a second inversion chord. The second inversion chord would be
iii6/4. To the extent that chord runs contrary to the dominant tendency of the bass we could take an alternate view and say that chord is really an appoggiatura or some other non-chord tone. If that tone steps up or down, it will move to a tone of the dominant chord.
In my mind, I think of it like this: the bass generates the harmony, the treble decorates the harmony, and the simplest functional analysis is preferred over unusual progressions.
Can a song have more than one chord progression at different points in the song.
I don't think you mean polytonality or something like that, so yes a song can have more than one chord progression. Actually, that's what gives good structure to the sections of songs. Contrasting progressions distinguishes sections. Also, a phrase of music can be re-harmonized so that the melody is kept the same and repeated but the accompanying chords are changed.