Sir Paul's technique is typical for what you might see in a self-taught player who began on guitar. I would not use the technique he is demonstrating in that video as an example to teach yourself to play. It works for him, but there are other ways of playing that are superior in terms of ergonomics and efficiency, IMHO.
To answer your specific question, at arena volumes or when recording in the studio, it is necessary to mute every string of the instrument. If Paul used a different right-hand technique, such as the "floating thumb" right-hand technique that is currently very popular on TalkBass, he would not need to use his left-hand thumb to mute the E and/or A strings.
I would guess that Paul developed his technique organically, developing new habits and methods over time as the Beatles grew. For example, when they moved to playing louder amplified music, that is when he would have been forced to come up with an effective string-muting strategy. Probably he just took what most electric guitar players of the time were doing with their left thumbs, and moved it over to bass, since there weren't really any "electric bass 101" methods back then to guide him (and he was too busy making lots of money).
(edit: Paul is left-handed, so for him, it's the opposite of what I said above.