One good way of ear training is to listen to music, and play each note along the low E string. You're looking for the note that sounds like the home note or the most resolved note. That'll tell you the key, then ask does the song sound happy or sad. Then you'll know the key like G Major, F minor. It'll take a little time. But when you figure out the key, you can figure out each chord that's in that key.
I liked to teach interval training without attaching key, or even telling notes. I'd rather be able to play 2 notes without telling you anything and have you know what the interval is. That way, you can find your way around by ear without having to think about the chord structures. In music school we learned in parallel but separately from written theory. Allows you to combine the two in your head at your own rate, and doesn't marry you to an instrument for training. A major 6th sounds functionally the same regardless of key, and most people know the NBC jingle, so it's easy to point that out. Or that the Simpsons opens with a tritone, or that the jaws theme is a minor 2nd, or Somewhere Over the Rainbow is an octave, etc etc.
Personally, I would get comfortable with melodic intervals before starting to tackle any serious chord identification. Not many people are going to come across, say, a half-diminished chord in popular music, but you're not going to recognize it if you can't identify the tritone in the middle.
Take all this with a grain of salt, everyone has their own method for sure, this is just how they did it at my school, which is different from the Paris Conservatory curriculum, which is different from some of the German takes (as a sax player the Sigurd Rascher school comes to mind, where as I came from the Larry Teal line by way of Fred Hemke,
Technically my teaching tree puts me at 5 degrees of separation from Adolph Sax).
As for figured bass, obviously it doesn't always work, but dear God is it amazing to see realized by an absolute pro. Our organ professor could play masses from memory but would always realize the figured bass differently, was truly incredible to watch.