Functional ear training, and hearing the degree instead of the chord
Hearing which degree a chord occupies in the key, and not only its quality; the skill that lets you accompany a song you have never heard, and the trick of singing the tonic while it plays.
Recognising that a chord is a minor seventh says what it is. It does not say where it is: whether it is the second degree preparing the dominant, or the sixth degree resting, or the third passing through. Functional ear training is hearing the degree, and it is the skill that lets you accompany a song you have never heard, because degrees repeat from song to song and chord names do not.
The tonic as the ruler
The central trick is simple and it works: sing the tonic while the music plays. Held in your head or in your voice, it is the ruler each chord is measured against, and the degree appears through the chord's relationship to it: the chord that rests on it is the I, the one that pulls to it is the V, the one that departs is the IV.
Sing the C and listen to the sequence. The F sounds like leaving home, the G seventh like wanting to come back, the C like arriving. You do not need to know the names: you feel the place.
Listening to the bass
The bass gives the degree away more often than any other note. A bass that leaps a fourth up has reached the subdominant, or the tonic coming from the dominant; a bass that rises a semitone to the tonic came from the leading tone. Whoever accompanies by ear listens to the bass first, decides the degree, and only then confirms the quality.
The basses of I, IV, V, I: it is the shape you learn to recognise, and it is the same in any key, because it is made of distances.
The degrees that get confused most
Three pairs fool people more than the rest, and they are worth knowing by name.
I and vi. They share two notes, and the sixth degree sounds like a tonic that has darkened. The bass decides: a minor third below the tonic is the vi.
IV and ii. They share two notes and the function. The ii is minor and the IV is major; and the ii's bass is a tone above the tonic, the IV's a fourth.
V and vii°. They share the tritone. The vii° has no dominant root in the bass and sounds tenser and thinner; in popular practice it appears rarely, and nearly every chord pulling to the tonic is the V.
Outside the field
A secondary dominant is heard as a V that pointed at the wrong place: the same wish to resolve, but towards a degree that is not the I. A modal borrowing is heard as a familiar degree that suddenly darkened, the minor IV being the commonest. In both cases the reference is still the sung tonic; what changes is that the chord arrived from outside, and the ruler says where from.
How to train it
This site's practice plays a short progression in a key and asks for the degrees; the options are progressions that get confused with one another. Listen to the bass, sing the tonic, decide each chord's function, and only then the degree. Away from the site, the exercise is accompanying unfamiliar songs on the radio: guessing the next chord before it arrives, and checking. Every hit is the skill forming; every miss says which pair still confuses you.
What to take from here
Functional ear training is hearing the degree, not the chord, and the ruler is the sung tonic. The bass gives the degree away more than any note; I and vi, IV and ii, V and vii° are the pairs that fool you; chords from outside are heard as familiar degrees displaced or darkened. It is the skill that makes an unfamiliar song playable on first hearing, and it is trained by accompanying.