That's a fantastic clip!...Brian at his peak - just before it all went a little awry.
Brian Wilson threw out the rulebook when recording in the mid 60s (Pet Sounds and Smile) and became a massive innovator - he pretty much ditched the band as studio musicians and used top session guys (The Wrecking Crew - Hal Blaine on drums, Carol Kay etc..)
He used bass harmonicas, theremins, harpsichords ,cellos, you name it - all to get the 'texture' right. Electric and upright bass played together would have been purely for texture and a 'different' sound. With Smile, given free reign by Capitol, he pretty much went mad (literally!) writing and recording small sections of exquisite music....just gave up in the end as stitching all the little bits together proved too much. I'd recommend getting the Smile sessions box set if you want to hear more of that - stunning!
I wouldn't say two basses used like that was a common technique - however, I'm certain Phil Spector will have done that. By the mid 60s electric bass ruled the roost in pop and soul. Some soul records that used big orchestras still had upright basses, and certainly some of the earlier doo woo inspired tracks.
Hopefully someone will give us some good examples of upright bass tracks...and even dual bass?