As detailed on the other thread,...
A lot of the "Beatles sound" came from the characteristic "limitations" and sound of the early vintage recording equipment.
Another part of the "Beatles sound" came from the recording techniques that were current at that time, which were early techniques developed in order to cope with the limitations of the early recording equipment.
The two, [vintage equipment/vintage technique], kind of go hand in hand.
More specifically, IMO:
1- The early 'Beatles-y' sound is a very 'midrange-y' sound. You don't typically have 'heavy kick drum' or 'thundering bass' in the early Beatles' recordings, like we're used to with today's recordings. This was indicative of the sound and limitations of the early Beatles-era recording equipment.
2- The early 'Beatles-y' sound is one which "packed the tracks", that is, put as many parts on each track as possible. This is also due to the limitations of early Beatles-era recording equipment, with many early Beatles recordings made direct-to-2-track, and later 3-tracks, then 4-tracks, as recording technology improved.
3- The early 'Beatles-y' sound is one which used ultra-wide, somewhat unusual PANNING of the tracks, f/i with all instruments panned to one side, and all voices panned to the other side. This, IMO, was due in part to the limitations of early Beatles-era recording equipment, and partly due to the relative NEWNESS of STEREO, and the efforts of recording engineers of the day to COME TO TERMS with this new thing called STEREO. Today, this type of extreme panning would seem somewhat 'unnatural' sounding, but it's a big part of the early 'Beatles-y' sound.
Later on, as recording equipment improved, the complexity and depth of sound of Beatles recordings also improved.