I was wondering whether it is possible to change any songs with the same chord progression into the same key?
Sure, in the chord progression used in the axis of 4 chords for example you have I V vi IV, or 1 5 6 4 if you will, in major.. easily transposable.
Listing the chords found within major:
I major
II minor
III minor
IV major
V major
VI minor
VII diminished (half diminished if you include the 7th)
So, constructing the chord progression in C major you would then play the C major (1, the tonic) G major (5, the dominant) A minor (6, the relative minor) and F major (4, the sub dominant).
Moving it to another key, for example F major would then be F major, C major, D minor and Bb major.