Yes, Sgt Pepper was recorded on a four track tape machine. The whole album...
Below is how they took four tracks and via bouncing some of them to another tape machine were able to record nine tracks of audio.
Trk 1.. Drums + Guitars
Trk 2.. Bass
Trk 3.. Lead + Harmony Vocals
Trk 4.. Additional Harmony Vocals
Recorder (B) -- The "BOUNCE" tape. This is what we're hearing in the video:
Trk 1 [GREEN ].. From A, Tracks 1+2 (drums, guitars, and bass)
Trk 2 [ BLUE ].. Horns and punched-in lead guitar
Trk 3 [ RED ].. From A, Tracks 3+4 (all vocals)
Trk 4 [YELLOW].. Sound effects