This is the "Pops Orchestra" or "pops" style, a pop-influenced style performed by a full symphonic orchestra. The repertoire is usually a mixture of famous classical music performed in a jazzy, pop style, or older popular music scored for orchestra. There's some original music specific to the genre, but that's comparatively rare. There's also some overlap with the poppy instrumental style known by the trademarked name "Muzak," aka "Elevator Music." When it's an actual pop song with vocals (as in your examples) that's technically called "chamber pop," but not all chamber pop has that distinctive '60s sound.
The genre was probably innovated, or at least popularized by the Boston Pops Orchestra, an offshoot of the Boston Symphony that is actually over 130 years old, but that really became popular and influential starting in the 1930s.
Speaking of which, while its hey-day was definitely in the 1950s and 60s, this genre didn't really go anywhere (except off the radio). Many large cities still have a live "Pops Orchestra," or at least a classical orchestra that occasionally plays in the "pops" style, particularly around the holidays --"pops" style Christmas music is still popular.
Here's my own favorite "pops orchestra" song, Paul Mauriat's immortal version of the French pop hit "Love is Blue":